Journal of Neuroscience Methods, 43 (1992) 201-214
201
© 1992 Elsevier Science Publishers B.V. All rights reserved 0165-0270/92/$05.00 NSM 01387
Modeling of the quantal release at interneuronal synapses: analysis of permissible values of model moments A.E. Dityatev, V.M. Kozhanov i and S.O.
Gapanovich
Laboratory of Evolution of Interneuronal Interaction, Sechenov Institute of Evolutionary Physiology and Biochemistry, Russian Academy of Sciences, St. Petersburg 194223, (Russia) (Received 2 March 1992) (Revised version received 13 May 1992) (Accepted 14 May 1992)
Key words: Q u a n t a l r e l e a s e m o d e l ; M e t h o d o f m o m e n t s ; F r o g spinal cord; S e n s o r i m o t o r synapse A theoretical study of effects of the different factors on fluctuation of post-synaptic potential (PSP) amplitudes was undertaken, using computation of regions of permissible values (RPV) of the ratio between the variance and the mean number of the quanta released (R 1) and the ratio between the third moment and the variance (R2). The RPVs of these indexes for the binomial model were compared with regions determined for a number of models incorporating several factors. It has been shown that the involvement of temporal non-uniformity of transmitter release probability, decremental spreading of potentials along dendrites, and failure of spike propagation give the values of skewness index R 2 less, compared to the binomial model. Simultaneously, a number of other factors, especially spatial non-uniformity of release probabilities in single release sites, would give amplitude histograms with high positive values of the index. The values of R I and R2, calculated for 21 samples of sensorimotor EPSP amplitudes, were biased from RPV of these parameters constructed for the binomial model. The scattergram of R x and R 2 can be explained by the presence of two kinds of contacts which release quantum with different probabilities. The same was true for the beta-model based on the assumption that probabilities of quantal release are a sample of values of random variable that has beta-distribution. From analysis of the distribution of individual release probabilities, obtained from evaluation of beta-model parameters, is concluded that a greater part of boutons in the sensorimotor synapses release transmitter with very low probabilities, there being, however, a few boutons with probabilities close to 1.
Introduction Originally, analysis o f f l u c t u a t i o n s o f post-synaptic p o t e n t i a l (PSP) a m p l i t u d e s at t h e i n t e r n e u r o n al synapses was b a s e d on t h e b i n o m i a l m o d e l (Katz an d Miledy, 1963; K u n o , 1964). This simple
i Present address: Neurobiologie Cellulaire, Institut Pasteur, 25 et 28, rue du Docteur-Roux, 75724 Paris Cedex 15, France.
Correspondence: A.E. Dityatev, Department of Physiology, University of Bern, Buhlplatz 5, CH-3012, Bern, Switzerland. E-mail: [email protected]; Tel.: 31-658-725; Fax: 31-654-611.
m o d e l is d e f i n e d by a p o p u l a t i o n of q u a n t a l units w h i ch f u n c t i o n i n d e p e n d e n t l y (i), with t h e s a m e probability (ii) an d the s a m e c o n t r i b u t i o n of e a c h to t h e r e c o r d e d p o t e n t i a l (iii). A r e m a r k a b l e o n e to-one relationship between binomial parameter n an d the n u m b e r of synaptic b o u t o n s ( K o r n et al., 1982; B a b a l i a n and C h m y k h o v a , 1987) was f o u n d using t h e b i n o m i a l m o d e l . S i m u l t an eo u sl y , d i s c r e p a n c i e s b e t w e e n b i n o m i a l and e x p e r i m e n t a l distributions o f P S P a m p l i t u d e s ( K o r n and Mallet, 1984; B a r t et al., 1 9 8 8 b ) w e r e r ev eal ed . T h e b i n o m i a l p a r a m e t e r n a p p e a r e d to be sm al l er t h a n t h e n u m b e r o f contacts f o r m i n g s e n s o r i m o t o r synapse o f frog and i n t e r n e u r o n a l synapses in
202 cell culture (Grantyn et ai., 1984b; Pun et al., 1986). The changes of n during post-tetanic potentiation, facilitation (for review see Redman, 1990) and alteration of extracellular calcium concentration (Dityatev et al., in preparation) are also in contrast with structural interpretation of this parameter. Progress was made by a deconvolution procedure (Edwards et al., 1976a, b; Wong and Redman, 1980) which allows elimination of the effects of contaminating noise and thus reveals a true pattern of PSP fluctuations without any serious restrictions on distribution of signal. However, if the ratio between the quantal size and noise standard deviation is less than 1.5, the estimation of the quantal size obtained by this procedure is dramatically biased (Clamann et al., 1991). The next step was to apply the unconstrained quantal model (Bart et al., 1988a; Walmsley et al., 1988) which is based on the assumption that a PSP fluctuation pattern consists of discrete components which are integer multiples of a unit increment. The model involves a smaller number of parameters and gives a more accurate estimate of the quantal size and probabilities of the occurrence of different quantal events (Dityatev, 1991). However, an unconstrained quantal model gives only limited information about the transmitter release mechanisms. From our point of view, evaluation of these model parameters should be considered to be an interim step. It can be used for evaluation of parameters of compound binomial quantal model (Walmsley et al., 1988), ie., for evaluation of release probabilities (Pl, Pz . . . . . PN) in single contacts. But search of the total number of quantal units ( N ) is a rather difficult task. This especially refers to large values of N which are typical, e.g., for frog sensorimotor synapse (Grantyn et al., 1984a) and synaptic connections between neurons in cell cultures (Neale et al., 1983). For such synapses, it is necessary to evaluate the parameters of the distribution of individual release probabilities (Pi) rather than individual values Pi. Convolution of two binomial distributions (Hatt and Smith, 1970; Bart et al., 1988b), the beta-model (Bennett and Lavidis, 1979; Zefirov and Stolov, 1982) and the model with sym-
metrical distribution of p~ (Myamoto, 1986) are all examples of models describing spatial non-uniformity of individual transmitter release probabilities. Except for the number of quantal units and average release probability, these models involve an additional parameter. After evaluation of the quantal size and distribution of the number of quanta released by an unconstrained quantal model, the above parameters can be estimated by the method of moments using the mean, the variance and the third moment of the revealed distribution. In this work, an attempt is made to develop an approach to estimate contributions of spatial non-uniformity of transmitter release probabilities, decremental spreading of PSPs along the dendrites, interaction of the neighboring contacts, and other factors to the fluctuation of PSP amplitudes. This approach is based on calculation of variation and skewness indexes (R l and R 2) of the revealed PSP amplitude histograms. The indexes are used to analyze the agreement between values of the indexes computed for the models and a set of experimental samples. This is similar to the analysis of a relationship between theoretical and experimental values of coefficient of variation used in the original work of Del Castillo and Katz (1954). The method is applied to study synaptic transmission in frog sensorimotor synapse.
Methods
Experimental data Isolated spinal cord of adult frogs Rana ridibunda was used in experiments. Animals were anesthetized by ether. After dorsal laminectomy, the spinal cord was removed and hemisected sagittally. Synaptic transmission was studied at the connections between Ia afferent fibers and spinal motoneurones of L9_|0 under intracellular (Grantyn et al., 1986b) and minimal extracellular stimulation of afferents. Morphological data used were obtained by staining of contact elements with horseradish peroxidase (Grantyn et al., 1986a; Chmykhova et al, 1987).
203
Data collection and implementation of models EPSPs were sampled on-line by a laboratory computer system (12-bit A-D converter, PDP 1 1 / 3 4 ) at the rate 12.5 kHz. Amplitude measurement was implemented under visual control by the method described by Yamamoto et al. (1991). Estimates of the quantal size were obtained by the maximum likelihood method as one of the unconstrained quantal model parameters (Bart et al., 1988a; Walmsley et al., 1988, Voronin et al., 1991). Implementation of the models was run on
an IBM P C / A T computer. The program was written in C and is available on request.
Binomial model Fluctuation of signal produced by a set of quantal units functioning according to the principles (i)-(iii) is described by a random variable (RV) 77 = Q" ~:, where Q is the quantal size (constant) and ~ is an RV which has binomial distribution with parameters p and n. This relation is denoted as ~ ~ / 3 + ( k i n , p) = Ck, pk (1 _ p ) , - k .
B
A
looI
0.80
~"2
--~2
J I/ /
0.60
R 2 = 2 . 2 5 R 1 + 0 . 2 5 / R 1 - 1.5./ / " / ' / / / / / I / 1/ /
0.40 R2=2Rl-1
0.20
O.O0
/
-0.20 /
-0.40 /
-0.60
/
/
/ /
/
- 1 .O0
I
,
,
I
,
I
/
/
/
/ / /
/
/ /
/
-0.80
/ / J
i
I
[i
I
I
C '°°f~
0.80 o.6o
.J
R =R
/
/(2-R
2\ 1
)
.2-t7:"
I
I)
/ J -~..;~'/~"
0 . 4 0 I - R 2 = ( l _ 3 R ' >,'_,O, values of R 2 will be smaller then 2 R j - 1 (binomial RPV) for an arbitrary choice of function describing the spreading of signals along dendrites.
Model with lateral interaction Suppose that binomial assumption (iii) is adopted but (i) and (ii) fail due to the fact that quantum release in some contact leads to a fast change of release probabilities at adjacent contacts (Shapovalov and Shiriaev, 1987). Then amplitude fluctuations are defined by RV
neighboring contacts has been proposed by Korn and Mallett (1984) to explain discrepancy between entropies of experimental and binomial distributions. Faber et al. (1985) demonstrated that the occlusion depends on overlapping of postsynaptic receptor zones. In case of occlusion fluctuation of PSP amplitude may be described by Eqn. 16. If contacts are arranged in line, the distances between them are equal and the contacts have equal effectiveness, then the distribution of responses from a group of adjacent contacts is defined by the set of recurrent equations:
P(rli= k . Q + c . l . Q ) =pk+l(1 _p)n,-k
t(Lk,/+ Fk,t ~ Hi
tl i
J
for k = 0,1,2 . . . . ,[ni/2]; l = 0,1,2 . . . . . n i - k
L~,t = F~-tl ,t + ~,-lk't-1l F f " = F f j , + L*#[ ,
G
r / = Q ~ r/i , i=1
(16)
FI ''°= 1, L', '°= 1, the others F~ j = L ] " = 0,
(is)
where G is the number of groups of adjacent contacts (contact zones) and r/i are RV described by the distributions /3+(. I ni, p, r). These depend on the number of contacts belonging to the ith z o n e (ni) , release probability before the first quantum release (p), and the extent of change of release probabilities (r). If release probabilities change r times after each release, probability of k quanta release is given by the following recurrent relation:
where c is the parameter defining the extent of overlapping of neighboring receptor zones (response from two boutons will be equal to Q + c • Q rather than to 2 " Q ) , n i is the number of contacts into ith group and Ix] is the least integer more than x.
The curves of R~ - R 2 dependence were built for this and the next model in the same way as for decremental spreading model.
Model of temporal variation of p Recent studies of synaptic transmission in hippocampal neurons suggest that the release probability of the mossy fiber terminal fluctuates temporally according to gamma-distribution (Yamamoto et al., 1991). Histogram of PSP amplitude for this case is described by Pascal distribution. Mean of this distribution is less than variance and this model does not provide estimation of the number of contacts. Therefore for a number of connections, including frog sensorimotor synapse, the binomial model with fluctuating p will likely be more appropriate. As our choice of distribution of p, we adopt a beta-function. Then probability of k quanta release is
Model with linear occlusion Existence of occlusion of contributions of
P ( ~ = k) = C ~ . B ( a + k, n + b - k ) / B ( a , b). (19)
n i -k
[3+(klni, P , r ) =
y" [ 3 + ( k - l l n i - t - 1
'
t = o
Fr(P), r ) - p ( 1 - p ) ' ]3+(0 [ni, p, r) = (1 - p ) " ' F~(p)=
rp, 1,
rp~l
(17)
207 100
Deriving the frequency-generating function (FGF), we have the expressions for moments:
0.80 060 0.40
E[~] = n a / ( a +b),
D[~]=nab(a+b+nl/[(a+b)Z(a+b+
02O
1)],
A -0.20
(20)
-0.40 -060
A[£] = n a b ( b - a)(a +b +n)(a +b + 2n)
-0,80 L
- 100
i
i
i
i
/ [ ( a + b)3(a +b+ l ) ( a +b+ 2)]. 1.00
The variance will be less than the expectation if
n < 1 + a + a/b +a2/b.
0.80 0.60
(21)
This condition will hold if the value of parameter a is sufficiently higher than that of b. Values of parameters R~ and R z for this model lie lower than the RPV for binomial model (Fig. 1A), that is in contrast to the beta-model with spatial nonuniformity of release probabilities.
0.40
]B
0.20 0.00 -0.20 -0.40 -0.60 -0.80
o8.~ o~
- 1.00
1.00
Variable quanta model Random fluctuation of the quantal size in the central synapses has been reported (Korn and Faber, 1990; Yamamoto et al., 1991). This factor can be described by Eqn. 14, where Qi are independent RVs rather than constants. If distribution of Qi is symmetrical with expectation E[Q] and variance D[Q], then moments of RV rt are given by expressions: E[r/] = E [ Q ] ' E [ ~ ] O[r/] = E 2 [ Q ] .D[~¢] + D [ Q ] "E[~]
0.60 0.40
C
0.20 0.00 -0.20 -0.40 -0.60 ~ 0~80
- 1.oo
'
0.00
'
0.20
'
0.40
,
0.~50
,
i
0.80
1.00
Fig. 2. R I -R 2 relationships for several value of quantum variance CVQ (A), probabilityof excitingof two fibers Pf (B) and drivingforce F (C).
A[~/] = E 3 [ Q ] "A[~:] + 3E[Q] "D[Q] " n [ ~ ] . (22) The curves indicating a change of parameters R1 and R 2 (p ranging from 0 to 1) under several fixed values of variation coefficient of quantal size are shown in Fig. 2A. They appeared to lie above the RPV for bonomial model, especially so for small values of R 1.
Model with convergent inputs It is possible that current impulses, injected through stimulating electrode, stimulate one fiber (with probability Pf), in addition, some other fiber is activates from time to time (with probability 1 - P f ) . If transmitter release agrees with the
binomial assumptions, then the distribution of the number of releasing quanta (~:) is a mixture of two binomial distributions: P(~:= k)
=Pf'fl÷(k In1, p) + (1 - P f ) •fl+(kln I +nz, p), k=0,1 . . . . ,n 1 +n2, (23) where n a and n 2 are the number of units activating by the both fibers, p is release probability. Fig. 2B demonstrates that values of R 2 can be high for arbitrary values of R~, growth of R 2
208
takes place with an increase of p (decrease of R~) above a certain threshold value.
r/, we derive FGF of r/ which is the composition of FGFs of O and ~ (Feller, 1984). The moments are given by expressions:
Failure model
E[r/] = Q . E [ ~ : ] "E[O]
Activation of a variable number of contact zones can be caused by the failure in spike propagation along collateral branches (Luscher, 1990). For a simple case the involvement of this factor is described by the RV:
D[r/] = Q2(O[~C] "E[O] +O[O] "E2[sc]) A[r/]=Q3(A[~].E[O]+E3[~
+ 3 E [ sc ] • D[~:]. D [ O ] ) .
o
r / = Q ~] ~:i,
c].A[O] (25)
Analytical investigation of expressions for R 1 and R 2 proves that these values are situated under binomial RPV, if sc and O are binomial RVs.
(24)
i-1
where O is RV modeling the number of contact zones at which spike has been arrived. Suppose that transmitter release in the zones follows the same rule (~:). In order to obtain moments of RV
Modeling of non-linear summation Taking into consideration the non-linearity between amplitude of PSP and conductance (Martin,
-5700 I
B
-5720 "~
-5740
O ,t= •-
-5760
\
o
-5780 O
-5800 50
i
r
i
70
90
110
\
\
i
130
150
Quantal s i z e (j~V) 50
/
40
8-1
20 10 0 -0.05
0.5 EPSP amplitude (mV)
Fig. 3. A: time courses of smoothed individual and averaged sensorimotor EPSPs. B: dependence of logarithm of the likelihood function on the quantal size. Optimal value of the size (Q) is 70/xV. C: averaged EPSPs with amplitude in the range Q +_ 1 / 2 • Q, 2 • Q _+ 1 / 2 • Q , . . . , 7- Q _+ 1 / 2 • Q. D: histogram of amplitude and fitted density of unconstrained quantal model.
209 1.00
1977), the revealed potential v is related to the number of released quanta (k) by
v
=
kQ . F / ( k Q
+
F),
0.80 0.60 0.40
(26)
where F is the driving force. Supposing that the number of released quanta is described by binomial RV, negligible increase of R 2 over simple binomial model RPV has been found (Fig. 2C).
•
0.20
A
0.00
•
.-'-~"° • //
/~
1/
•
2
/
-0.20
l////~f
-0.40
/I/
-0.60
.
-0.80
,
////
,
,
i
i
i
i
e
- 1.00
1.00
Results
0.80
R2
0.60
Comparison of experimental and theoretical values of R 1 and R 2 Twenty-one Ia a f f e r e n t - m o t o n e u r o n connections were taken for statistical analysis. Individual EPSPs and steps of evaluation of the quantal size are shown in Fig. 3. Only for 3 amplitude samples, the values of R 1 and R 2 were situated lower than RPV for the binomial model (Fig. 4A). If EPSP fluctuation were described by binomial model, then about half of the points would fall below the line (Fig. 4B). This discrepancy is significant ( P < 0.01; chi-square test). The rest 18 points lay inside the binomial convolution RPV. Most of the experimental points form a cluster with the center inside beta-model RPV (Fig. 4A). The cluster is farther from RPV for Myamoto's model than from the beta-model region. It seems plausible that the scatter diagram observed is a result of statistical fluctuation of R 1 and R2 values. To verify this suggestion we simulated 20 samples of PSP amplitudes described by betamodel with values of parameters a = 0.17 and b = 0.68. These values were obtained from the expressions 10 and 11, where RI and R 2 were the coordinates of cluster center. The scattergram obtained after simulation (Fig. 4C) turned out to be similar to the experimental one. This leads to a hypothesis that Pi distributions at the connections under study can be described by a common law. Regions of permissible values of models incorporating decremental spreading, lateral interaction and occlusion are presented in Fig. 5. The sets of curves were drawn for the connection shown in Fig. 3 (Grantyn et al., 1984b). Evidently, only the model with lateral interaction gives val-
0.40
~t
0.20
B
0.00 g•
-0.20
•
-0.40 -0.60 -0.80 - 1.00
1.00 0.80 0.60 0.40 O2.0 C
0.00
0.20
, t...~
-0.40
-0.60
-o.ao -
1.00
0.00
. .~
.-/
•
/
R, i
0.20
~
040
o.eo
o.ao
e
1.00
Fig. 4. Scatter diagram of values R s and R 2 calculated for 21 samples of frog sensorimotor EPSP (A) and simulated values of R 1 and R 2 for binomial model (B) and beta-model (C). + is the center of points. Parameters: binomial model, p = 0.6, n = 15; beta-model, a = 0.17, b = 0.68, N = 45. Sample size was 400.
ues of R 1 and R 2 situated above the RPV for binomial model. This is a case in which the values of parameter r are negative, i.e., under lateral inhibition. However, even then the experimental point is situated above RPV. Shapes of the curves for 4 other connections were very similar to the above and the revealed relationships between binomial RPV and values of R 1 and RE calculated for these three models were presented.
210 TABLE 1 ESTIMATES OF T H E PARAMETERS OF T H R E E Q U A N T A L RELEASE MODELS A N D M O R P H O L O G I C A L PARAMETERS OF F R O G S E N S O R I M O T O R CONNECTIONS No.
Quantal size
Noise SD
Binomial model
Binomial convolution model
Betamodel
(/..tV)
(#V)
n
p
NI
Pt
N2
Nt + N2
N
M0
Number Bouton
Contact zone
40 43 42 72 26
11 14 15 23 8
1 2 3 4 5
67 57 70 92 60
30 38 29 48 30
16 22 16 20 9
0.63 0.38 0.66 0.72 0.60
17 29 14 18 9
(I.29 0.26 0.44 0.36 0.49
5 1 4 8 1
22 30 18 26 10
50 43 52 74 26
6 3 4 9 2
6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
109 73 71 71 79 61 57 122 80 91 61 61 91 76 81 62
43 27 33 77 41 36 57 52 35 40 56 77 36 37 28 32
17 17 15 37 6 14 16 11 22 13 16 8 3 13 7 12
0.81 0.71 0.70 0.39 0.63 0.47 0.54 0.60 0.40 0.62 0.28 0.66 0.69 0.61 0.58 0.51
10 17 15 . 7 15 16 11 32 13 . 7 3 16 7 13
0.54 0.25 0.27 . . 0.35 0.39 0.40 0.62 0.20 0.55 . . 0.58 0.69 0.33 0.44 0.31
8 8 6 . 1 1 2 0 2 1 . 1 0 3 1 2
18 25 21
10 8 6 8 I 1 2 3 4 3 1 1 0 4 1 2
Have not measured
8 3 19 8 15
69 62 51 74 17 34 43 33 43 42 23 25 11 41 20 31
42 14.6
13 5.1
0.60 0.09
14 6.9
0.41 0.13
17 7.9
40 15.1
3.7 2.4
45 16.8
models
are
Mean SD
76 17.3
Estimates o f quantal release parameters Evaluated
parameters
given in T a b l e
I. T h e
of
mean
three number
8 16 18 11 34 14
2.9 2.4
units of quantal
times
predicted the
by beta-model
binomial
(N)
parameter
differ from the number
14 5.6
n
of boutons
exceeded and (t
did
3 not
test, P>
TABLE II ESTIMATION OF C O N T R I B U T I O N OF SEVERAL FACTORS BY Q U A N T A L ANALYSIS OF THE F L U C T U A T I O N OF PSPs Factor
Skewness test a
Morphological test b
Spatial non-uniformity Pi Temporal non-uniformity p Variation of quantal size Decrement spreading of local PSP Lateral inhibition Occlusion Non-linear summation Failure
+4-
+
+ + + + -
+ + + -
~' Two crosses: agreement between theoretical and experimental values of variation and skewness indexes; one cross: values of skewness index are more than binomial one; minus: reverse relationship. b Cross: the number of quantal units predicted by binomial model (n) is smaller than real value; minus: reverse relationship.
211
0.1). Correlation coefficient between the number of boutons and N was 0.95 (5 connections). Binomial parameter n was close to the numbers of contact zones (correlation coefficient cor. = 0.84) and to the number of units functioning with probabilities Pi > 0.2 (cor. = 0.85). The latter number was predicted by beta-model. The estimate of mean probability of quantum release obtained by beta-model was approximately 0.2. The mean number of the quantal units estimated by binomial convolution model ( N 1 + N 2) was more than binomial n (17 vs. 13), being however much lower than that of boutons (45).
A
x
=
2
S
0
a 0 0.00
0.20
i
I
h
0.40
0.60
0.80
1.00
Release probability
1000;,500;300;~
The number of high-effective units predicted by binomial convolution ( N 2) were close to the minimal values of EPSP amplitudes divided by quantal size (M0). Fig. 6 represents shape of the distribution of release probabilities estimated by the beta-model. This curve corresponds to mean values of R~ and R 2 (cross in Fig. 4A and C) and gives integrative representation of the non-uniformity of the release probabilities at frog sensorimotor synapse. The density has two maxima at 0 and 1, the first being much expressive. The shape of the density curve agrees very well with the fact that the estimated number of high-effective units ( N 2) is much smaller than the total number of quantal units. The value of N 2 corresponds to the number o f quantal u n i t s w i t h Pi > 0.8 ( c o l = 0.8, 5 experiments).
- 1.00
i
=
. , .;.;;
.
O.6O
0.40 ÷ O.2O
B
I
Fig. 6. Estimated density of release probability in single boutons of frog sensorimotor connection (a = 0.17, b = 0.68).
tOO p~ 0.80 0.60 0.40 0.20 0.00 -0.20 -0.40 -0.60 -0.80
tOO 0.80
8
0.00 -0.20 -0.40 -0.60 -0.80 - 1.00
tOO c
0.80
C
=
0
.
3
~
0.60 0.40 0.20 0.00
Discussion
-020
--0.40 --0.60 --0.80 - 1.0,£
0.00
0.20
040
I
L
I
0.60
0.80
1.OO
Fig. 5. R 1 - R z relationships for several values of the electrotonic length ,~ (A), the interaction coefficient r (B) and the occlusion coefficient c (C). + is the value of R s and R 2 calculated for experimental sample of EPSP amplitudes.
Overestimation of the quantal size obtained by the maximum likelihood method (deconvolution procedure) as a parameter of model of mixture of normal distributions has been reported (Clamann et al., 1991). In the present study the unconstrained quantal model was used. The number of parameters to be estimated was nearly 2 times less, compared to the former model. Moreover, no correlation of quantal size and noise SD was
212
found (cor. = -0.011). This is in contrast to the results obtained under the study of synaptic transmission at cat interneuronal connections by several workers (Clamman et al., 1991). Moreover, if true values of quanta were less than those used for calculation of R 1 and R2, true points would be situated along the line connecting estimated values with the coordinate center and farther on the center than shown on Fig. 4A. Such displacement, however, would not be essential for the results presented here. It may be possible that a stimulating electrode activated a few rather than one afferent fibers. This could be the cause of high values of the asymmetry coefficient R 2 (Fig. 2B). However, this is unlikely because the amplitude of electrical component of frog single-fiber sensorimotor EPSP did not fluctuate (Grantyn et al., 1984b). The mean value of R 2 calculated under extra- and intracellular stimulations did not different statistically. Table II summarizes the results obtained in the study of the influence of several factors on the values of skewness index and on the number of quantal units predicted by binomial model. We conclude that only involving of spatial non-uniformity it is possible to account for the discrepancies of binomial and experimental values. On the other hand, it is not to be excluded that lateral inhibition of neighboring contacts and variation of the quantal size also contribute to fluctuation of frog sensorimotor EPSPs. The model of the convolution of two binomial distributions was sufficient to describe the scattergram of experimental values of R~ and R 2, but the predicted release sites were much less numerous than the observed. In contrast to this model, the beta-model gives a good explanation of the obtained scattergram and accurately predicts the number of contacts. Addition to the beta-model of the pool of contacts in which each impulse induces transmitter release gives RPV with high values of R 2. It can be expected for interneuronal connection of cat where high-effective contacts were revealed (Walmsley et al., 1988). Beta-model gives accurate interpretation of binomial n as the number of contacts in which quantal release occurs with probability of more than 0.2. Hence, considerable
underestimation of the number of contacts by binomial n is due to involvement in synaptic transmission of a large number of low-effective (Pi < 0.2), contacts which are responsible for the peak at 0 on Fig. 6. List of abbreviations
EPSP PSP RPV A a
B(a, b) b c D d E F G
M3 N Nl, N2
Pl, P2
ff p
Pi Q Qi Ri R2
S2
excitatory post-synaptic potential post-synaptic potential region of permissible values third central moment p a r a m e t e r of beta-distribution beta-function parameter of beta-distribution p a r a m e t e r defining extent of occlusion variance, dispersion part of quantal units functioning with
pi = 1 expectation driving force number of contact zones estimated third moment of the number of quanta released total number of non-uniform quantal units numbers of less and more effective quantal units binomial parameter, the total number of uniform units release probabilities of less and more effective units probability of single-fiber activation binomial parameter, release probability of uniform units mean release probability release probability of ith quantal unit quantal size contribution of ith quantal unit to recorded potential ratio between the variance and mean number of quanta released ratio between the third moment and variance of quanta released degree of interaction of adjacent contacts estimated variance of the number of quanta released
213
~7
mean number of quanta released length constant number of quanta released amplitude of recovered distribution of PSP
Acknowledgements
We would like to thank Dr. N.M. Chmykhova for morphological data, Prof. P. Clamann for critical reading and helpful comment on the manuscript, I.I. Evdokimov and A.B. Veremiev for programming assistance.
References Babalian, A.L. and Chmykhova, N.M. (1987) Morphophysiological description of connections between single ventrolateral tract fibers and individual motoneurones in frog spinal cord, Brain Res., 407: 394-397. Bart, A.G., Dityatev, A.E. and Kozhanov, V.M. (1988a) Quantal analysis of the postsynaptic potentials in the interneuronal synapses: recovery of a signal from the noise, Neirofiziologiia (Kiev), 20: 479-487. Bart, A.G., Dityatev, A.E. and Kozhanov, V.M. (1988b) Analysis of the transmission in the interneuronal synapses by the convolution of binomial distributions, Neirofiziologiia (Kiev), 20: 487-494. Bennett, M.R. and Lavidis, N.A. (1979) The effect of calcium ions on the secretion of quanta evoked by impulse at nerve terminal release sites, J. Gen. Physiol., 74: 429-456. Chmykhova, N.M., Shapovalov, A.I., Shiraev, B.I., Komissarchik, J.J. and Snigirevskaja E.S. (1987) Ultrastructural analysis of physiologically identified and stained with horseradish peroxidase the sensorimotor synapse of frog, Doklady AN SSSR (Physiol.), 293: 249-252. Clamann, H.P., Rioult-Pedotti, M.S. and Luscher, H.R. (1991) The influence of noise on quantal EPSP size obtained by deconvolution in spinal motoneurones in the cat, J. Neurophysiol., 65: 67-75. Del Castillo, J. and Katz, B. (1954) Quantal components of the end-plate potential, J. Physiol., 124: 560-573. Dityatev, A.E. (1991) Analysis of non-uniformity of transmitter release probabilities in frog interneuronal synapses. Ph.D. Thesis, Sechenov Institute of Evolutionary Physiology and Biochemistry, 186 pp. Dityatev, A.E., Gapanovich, S.O. and Kozhanov, V.M. Influence of extracellular Ca 2+ on transmitter release parameters at Ia-afferent fiber connections on motoneurones of frog (submitted to Neuroscience).
Edwards, F.R., Redman, S.J. and Walmsley, B. (1976a) Statistical fluctuation in charge transfer at Ia synapses on spinal motoneurones, J. Physiol., 259: 665-688. Edwards, F.P., Redman, S.J. and Walmsley, B. (1976b) Nonquantal fluctuations and transmission failures in change transfer at Ia synapses at spinal motoneurones, J. Physiol., 259: 689-714. Faber, D.S., Funch, P.G. and Korn, H. (1985) Evidence that receptors mediating central synaptic potentials extend beyond the postsynaptic density, Proc. Natl.Acad. Sci. USA, 82: 3504-3508. Feller, W. (1984) An introduction to probability theory and its applications, 3rd edn., Vol. I, John Wiley, New York, 527 PP. Grantyn, R., Shapovalov, A.I. and Shiriaev, B.I. (1984a) Tracing of frog sensory-motor synapses by intracellular injection of horseradish peroxidase, J. Physiol., 349: 441-458. Grantyn, R., Shapovalov, A.I. and Shiriaev, B.I. (1984b) Relation between structural and release parameters at the frog sensory-motor synapses, J. Physiol., 349: 459-474. Hatt, H. and Smith, D.O. (1970) Non-uniform probabilities of quantal release at the crayfish neuromuscular junction, J. Physiol., 259: 395-404. Katz, B. and Miledi, K. (1963) A study of spontaneous miniature potentials in spinal motoneurones, J. Physiol., 168: 389-422. Korn, H. and Mallet, A. (1984) Transformation of binomial input by the postsynaptic membrane at a central synapse, Science, 225: 1157-1159. Korn, H. and Faber, D.S. (1990) Transmission at a central inhibitory synapse. IX. Quantal structure of synaptic noise, J. Neurophysiol., 63: 198-222. Korn, H., Mallet, A., Triller, A. and Faber, D.S. (1982) Transmission at a central inhibitory synapses. II. Quantal description of release, with a physical correlate for binomial n, J. Neurophysiol., 48: 679-707. Kuno, M. (1964) Quantal components of excitatory synaptic potentials in spinal motoneurones, J. Physiol., 175: 81-99. Kuno, M. and Weakly, J.N. (1972) Quantal components of the inhibitory synaptic potential in spinal motoneurones of the cat, J. Physiol., 224: 287-303. Luscher, H.R. (1990) Transmission failure and its relief in the spinal monosynaptic reflex arc. In: M.D. Binder and L.M. Mendell (Eds.), The Segmental Motor System, Oxford University Press, New York, pp. 328-348. Martin, A.R. (1977) Junctional transmission. II. Presynaptic mechanisms. In: E.R. Kandel (Ed.), Handbook of Physiology, Sect. 1, Vol. 1, Pt. 1, Baltimore, MO, pp. 329-355. Miyamoto, M.D. (1986) Probability of quantal transmitter release from nerve terminals: theoretical consideration in the determination of spatial variation, J. Theor. Biol., 123: 298-304. Neale, E.A., Nelson, P.G. and McDonald, R.L. (1983) Synaptic interactions between mammalian central neurons in cell culture. III. Morphophysiological correlates of quantal synaptic transmission, J. Neurophysiol., 49: 1459-1468. Pun, R.Y.K., Neale, E.A., Guthrie, P.B. and Nelson, P.G.
214 (1986) Active and inactive central synapses in cell culture, J. Neurophysiol., 56: 1242-1256. Redman, S. (1990) Quantal analysis of synaptic potentials in neurons of central nervous system. Physiol. Rev., 70: 165198. Shapovalov, A.I., Shiriaev, B.I. (1987) Signals Transfer in Interneuronal Synapses, Nauka, Leningrad, 173 pp. Tuckwell, H.C. (1988) Introduction to theoretical neurobiology. Vol. 1. Linear cable theory and dendritic structure, Cambridge University Press, Cambridge, 291 pp. Voronin, L.L., Kuhnt, U. and Gusev, A.G. (1991) Analysis of EPSP fluctuations indicated increased presynaptic release during long-term potentiation in area CA1 of hippocampal slices, Neurosci. Res. Comm., 8: 87-94. Walmsley, B., Edwards, F.P. and Tracey, D.J. (1988) Nonuniform release probabilities underlie quantal synaptic trans-
mission at a mammalian excitatory central synapse, J. Neurophysiol., 60: 889-908. Wong, K. and Redman, S.J. (1980) The recovery of a random variable from a noisy record with application to the study of fluctuations in synaptic potentials, J. Neurosci. Methods, 2: 389-409. Yamamoto, C., Higashima, M., Sawada, S. and Kamiya, H. (1991) Quantal components of the synaptic potential induced in hippocampal neurons by activation of granule cells, and the effect of 2-amino-4-phosphonobutiric acid, Hippocampus, 1: 93-106. Zefirov, A.L. and Stolov, E.L. (1982) Model of transmitter secretion in neuromuscular synapse based on the spatial non-uniformity of release probability of acetilholin quanta, Neirofiziologiia (Kiev) 14: 233-240.
201
© 1992 Elsevier Science Publishers B.V. All rights reserved 0165-0270/92/$05.00 NSM 01387
Modeling of the quantal release at interneuronal synapses: analysis of permissible values of model moments A.E. Dityatev, V.M. Kozhanov i and S.O.
Gapanovich
Laboratory of Evolution of Interneuronal Interaction, Sechenov Institute of Evolutionary Physiology and Biochemistry, Russian Academy of Sciences, St. Petersburg 194223, (Russia) (Received 2 March 1992) (Revised version received 13 May 1992) (Accepted 14 May 1992)
Key words: Q u a n t a l r e l e a s e m o d e l ; M e t h o d o f m o m e n t s ; F r o g spinal cord; S e n s o r i m o t o r synapse A theoretical study of effects of the different factors on fluctuation of post-synaptic potential (PSP) amplitudes was undertaken, using computation of regions of permissible values (RPV) of the ratio between the variance and the mean number of the quanta released (R 1) and the ratio between the third moment and the variance (R2). The RPVs of these indexes for the binomial model were compared with regions determined for a number of models incorporating several factors. It has been shown that the involvement of temporal non-uniformity of transmitter release probability, decremental spreading of potentials along dendrites, and failure of spike propagation give the values of skewness index R 2 less, compared to the binomial model. Simultaneously, a number of other factors, especially spatial non-uniformity of release probabilities in single release sites, would give amplitude histograms with high positive values of the index. The values of R I and R2, calculated for 21 samples of sensorimotor EPSP amplitudes, were biased from RPV of these parameters constructed for the binomial model. The scattergram of R x and R 2 can be explained by the presence of two kinds of contacts which release quantum with different probabilities. The same was true for the beta-model based on the assumption that probabilities of quantal release are a sample of values of random variable that has beta-distribution. From analysis of the distribution of individual release probabilities, obtained from evaluation of beta-model parameters, is concluded that a greater part of boutons in the sensorimotor synapses release transmitter with very low probabilities, there being, however, a few boutons with probabilities close to 1.
Introduction Originally, analysis o f f l u c t u a t i o n s o f post-synaptic p o t e n t i a l (PSP) a m p l i t u d e s at t h e i n t e r n e u r o n al synapses was b a s e d on t h e b i n o m i a l m o d e l (Katz an d Miledy, 1963; K u n o , 1964). This simple
i Present address: Neurobiologie Cellulaire, Institut Pasteur, 25 et 28, rue du Docteur-Roux, 75724 Paris Cedex 15, France.
Correspondence: A.E. Dityatev, Department of Physiology, University of Bern, Buhlplatz 5, CH-3012, Bern, Switzerland. E-mail: [email protected]; Tel.: 31-658-725; Fax: 31-654-611.
m o d e l is d e f i n e d by a p o p u l a t i o n of q u a n t a l units w h i ch f u n c t i o n i n d e p e n d e n t l y (i), with t h e s a m e probability (ii) an d the s a m e c o n t r i b u t i o n of e a c h to t h e r e c o r d e d p o t e n t i a l (iii). A r e m a r k a b l e o n e to-one relationship between binomial parameter n an d the n u m b e r of synaptic b o u t o n s ( K o r n et al., 1982; B a b a l i a n and C h m y k h o v a , 1987) was f o u n d using t h e b i n o m i a l m o d e l . S i m u l t an eo u sl y , d i s c r e p a n c i e s b e t w e e n b i n o m i a l and e x p e r i m e n t a l distributions o f P S P a m p l i t u d e s ( K o r n and Mallet, 1984; B a r t et al., 1 9 8 8 b ) w e r e r ev eal ed . T h e b i n o m i a l p a r a m e t e r n a p p e a r e d to be sm al l er t h a n t h e n u m b e r o f contacts f o r m i n g s e n s o r i m o t o r synapse o f frog and i n t e r n e u r o n a l synapses in
202 cell culture (Grantyn et ai., 1984b; Pun et al., 1986). The changes of n during post-tetanic potentiation, facilitation (for review see Redman, 1990) and alteration of extracellular calcium concentration (Dityatev et al., in preparation) are also in contrast with structural interpretation of this parameter. Progress was made by a deconvolution procedure (Edwards et al., 1976a, b; Wong and Redman, 1980) which allows elimination of the effects of contaminating noise and thus reveals a true pattern of PSP fluctuations without any serious restrictions on distribution of signal. However, if the ratio between the quantal size and noise standard deviation is less than 1.5, the estimation of the quantal size obtained by this procedure is dramatically biased (Clamann et al., 1991). The next step was to apply the unconstrained quantal model (Bart et al., 1988a; Walmsley et al., 1988) which is based on the assumption that a PSP fluctuation pattern consists of discrete components which are integer multiples of a unit increment. The model involves a smaller number of parameters and gives a more accurate estimate of the quantal size and probabilities of the occurrence of different quantal events (Dityatev, 1991). However, an unconstrained quantal model gives only limited information about the transmitter release mechanisms. From our point of view, evaluation of these model parameters should be considered to be an interim step. It can be used for evaluation of parameters of compound binomial quantal model (Walmsley et al., 1988), ie., for evaluation of release probabilities (Pl, Pz . . . . . PN) in single contacts. But search of the total number of quantal units ( N ) is a rather difficult task. This especially refers to large values of N which are typical, e.g., for frog sensorimotor synapse (Grantyn et al., 1984a) and synaptic connections between neurons in cell cultures (Neale et al., 1983). For such synapses, it is necessary to evaluate the parameters of the distribution of individual release probabilities (Pi) rather than individual values Pi. Convolution of two binomial distributions (Hatt and Smith, 1970; Bart et al., 1988b), the beta-model (Bennett and Lavidis, 1979; Zefirov and Stolov, 1982) and the model with sym-
metrical distribution of p~ (Myamoto, 1986) are all examples of models describing spatial non-uniformity of individual transmitter release probabilities. Except for the number of quantal units and average release probability, these models involve an additional parameter. After evaluation of the quantal size and distribution of the number of quanta released by an unconstrained quantal model, the above parameters can be estimated by the method of moments using the mean, the variance and the third moment of the revealed distribution. In this work, an attempt is made to develop an approach to estimate contributions of spatial non-uniformity of transmitter release probabilities, decremental spreading of PSPs along the dendrites, interaction of the neighboring contacts, and other factors to the fluctuation of PSP amplitudes. This approach is based on calculation of variation and skewness indexes (R l and R 2) of the revealed PSP amplitude histograms. The indexes are used to analyze the agreement between values of the indexes computed for the models and a set of experimental samples. This is similar to the analysis of a relationship between theoretical and experimental values of coefficient of variation used in the original work of Del Castillo and Katz (1954). The method is applied to study synaptic transmission in frog sensorimotor synapse.
Methods
Experimental data Isolated spinal cord of adult frogs Rana ridibunda was used in experiments. Animals were anesthetized by ether. After dorsal laminectomy, the spinal cord was removed and hemisected sagittally. Synaptic transmission was studied at the connections between Ia afferent fibers and spinal motoneurones of L9_|0 under intracellular (Grantyn et al., 1986b) and minimal extracellular stimulation of afferents. Morphological data used were obtained by staining of contact elements with horseradish peroxidase (Grantyn et al., 1986a; Chmykhova et al, 1987).
203
Data collection and implementation of models EPSPs were sampled on-line by a laboratory computer system (12-bit A-D converter, PDP 1 1 / 3 4 ) at the rate 12.5 kHz. Amplitude measurement was implemented under visual control by the method described by Yamamoto et al. (1991). Estimates of the quantal size were obtained by the maximum likelihood method as one of the unconstrained quantal model parameters (Bart et al., 1988a; Walmsley et al., 1988, Voronin et al., 1991). Implementation of the models was run on
an IBM P C / A T computer. The program was written in C and is available on request.
Binomial model Fluctuation of signal produced by a set of quantal units functioning according to the principles (i)-(iii) is described by a random variable (RV) 77 = Q" ~:, where Q is the quantal size (constant) and ~ is an RV which has binomial distribution with parameters p and n. This relation is denoted as ~ ~ / 3 + ( k i n , p) = Ck, pk (1 _ p ) , - k .
B
A
looI
0.80
~"2
--~2
J I/ /
0.60
R 2 = 2 . 2 5 R 1 + 0 . 2 5 / R 1 - 1.5./ / " / ' / / / / / I / 1/ /
0.40 R2=2Rl-1
0.20
O.O0
/
-0.20 /
-0.40 /
-0.60
/
/
/ /
/
- 1 .O0
I
,
,
I
,
I
/
/
/
/ / /
/
/ /
/
-0.80
/ / J
i
I
[i
I
I
C '°°f~
0.80 o.6o
.J
R =R
/
/(2-R
2\ 1
)
.2-t7:"
I
I)
/ J -~..;~'/~"
0 . 4 0 I - R 2 = ( l _ 3 R ' >,'_,O, values of R 2 will be smaller then 2 R j - 1 (binomial RPV) for an arbitrary choice of function describing the spreading of signals along dendrites.
Model with lateral interaction Suppose that binomial assumption (iii) is adopted but (i) and (ii) fail due to the fact that quantum release in some contact leads to a fast change of release probabilities at adjacent contacts (Shapovalov and Shiriaev, 1987). Then amplitude fluctuations are defined by RV
neighboring contacts has been proposed by Korn and Mallett (1984) to explain discrepancy between entropies of experimental and binomial distributions. Faber et al. (1985) demonstrated that the occlusion depends on overlapping of postsynaptic receptor zones. In case of occlusion fluctuation of PSP amplitude may be described by Eqn. 16. If contacts are arranged in line, the distances between them are equal and the contacts have equal effectiveness, then the distribution of responses from a group of adjacent contacts is defined by the set of recurrent equations:
P(rli= k . Q + c . l . Q ) =pk+l(1 _p)n,-k
t(Lk,/+ Fk,t ~ Hi
tl i
J
for k = 0,1,2 . . . . ,[ni/2]; l = 0,1,2 . . . . . n i - k
L~,t = F~-tl ,t + ~,-lk't-1l F f " = F f j , + L*#[ ,
G
r / = Q ~ r/i , i=1
(16)
FI ''°= 1, L', '°= 1, the others F~ j = L ] " = 0,
(is)
where G is the number of groups of adjacent contacts (contact zones) and r/i are RV described by the distributions /3+(. I ni, p, r). These depend on the number of contacts belonging to the ith z o n e (ni) , release probability before the first quantum release (p), and the extent of change of release probabilities (r). If release probabilities change r times after each release, probability of k quanta release is given by the following recurrent relation:
where c is the parameter defining the extent of overlapping of neighboring receptor zones (response from two boutons will be equal to Q + c • Q rather than to 2 " Q ) , n i is the number of contacts into ith group and Ix] is the least integer more than x.
The curves of R~ - R 2 dependence were built for this and the next model in the same way as for decremental spreading model.
Model of temporal variation of p Recent studies of synaptic transmission in hippocampal neurons suggest that the release probability of the mossy fiber terminal fluctuates temporally according to gamma-distribution (Yamamoto et al., 1991). Histogram of PSP amplitude for this case is described by Pascal distribution. Mean of this distribution is less than variance and this model does not provide estimation of the number of contacts. Therefore for a number of connections, including frog sensorimotor synapse, the binomial model with fluctuating p will likely be more appropriate. As our choice of distribution of p, we adopt a beta-function. Then probability of k quanta release is
Model with linear occlusion Existence of occlusion of contributions of
P ( ~ = k) = C ~ . B ( a + k, n + b - k ) / B ( a , b). (19)
n i -k
[3+(klni, P , r ) =
y" [ 3 + ( k - l l n i - t - 1
'
t = o
Fr(P), r ) - p ( 1 - p ) ' ]3+(0 [ni, p, r) = (1 - p ) " ' F~(p)=
rp, 1,
rp~l
(17)
207 100
Deriving the frequency-generating function (FGF), we have the expressions for moments:
0.80 060 0.40
E[~] = n a / ( a +b),
D[~]=nab(a+b+nl/[(a+b)Z(a+b+
02O
1)],
A -0.20
(20)
-0.40 -060
A[£] = n a b ( b - a)(a +b +n)(a +b + 2n)
-0,80 L
- 100
i
i
i
i
/ [ ( a + b)3(a +b+ l ) ( a +b+ 2)]. 1.00
The variance will be less than the expectation if
n < 1 + a + a/b +a2/b.
0.80 0.60
(21)
This condition will hold if the value of parameter a is sufficiently higher than that of b. Values of parameters R~ and R z for this model lie lower than the RPV for binomial model (Fig. 1A), that is in contrast to the beta-model with spatial nonuniformity of release probabilities.
0.40
]B
0.20 0.00 -0.20 -0.40 -0.60 -0.80
o8.~ o~
- 1.00
1.00
Variable quanta model Random fluctuation of the quantal size in the central synapses has been reported (Korn and Faber, 1990; Yamamoto et al., 1991). This factor can be described by Eqn. 14, where Qi are independent RVs rather than constants. If distribution of Qi is symmetrical with expectation E[Q] and variance D[Q], then moments of RV rt are given by expressions: E[r/] = E [ Q ] ' E [ ~ ] O[r/] = E 2 [ Q ] .D[~¢] + D [ Q ] "E[~]
0.60 0.40
C
0.20 0.00 -0.20 -0.40 -0.60 ~ 0~80
- 1.oo
'
0.00
'
0.20
'
0.40
,
0.~50
,
i
0.80
1.00
Fig. 2. R I -R 2 relationships for several value of quantum variance CVQ (A), probabilityof excitingof two fibers Pf (B) and drivingforce F (C).
A[~/] = E 3 [ Q ] "A[~:] + 3E[Q] "D[Q] " n [ ~ ] . (22) The curves indicating a change of parameters R1 and R 2 (p ranging from 0 to 1) under several fixed values of variation coefficient of quantal size are shown in Fig. 2A. They appeared to lie above the RPV for bonomial model, especially so for small values of R 1.
Model with convergent inputs It is possible that current impulses, injected through stimulating electrode, stimulate one fiber (with probability Pf), in addition, some other fiber is activates from time to time (with probability 1 - P f ) . If transmitter release agrees with the
binomial assumptions, then the distribution of the number of releasing quanta (~:) is a mixture of two binomial distributions: P(~:= k)
=Pf'fl÷(k In1, p) + (1 - P f ) •fl+(kln I +nz, p), k=0,1 . . . . ,n 1 +n2, (23) where n a and n 2 are the number of units activating by the both fibers, p is release probability. Fig. 2B demonstrates that values of R 2 can be high for arbitrary values of R~, growth of R 2
208
takes place with an increase of p (decrease of R~) above a certain threshold value.
r/, we derive FGF of r/ which is the composition of FGFs of O and ~ (Feller, 1984). The moments are given by expressions:
Failure model
E[r/] = Q . E [ ~ : ] "E[O]
Activation of a variable number of contact zones can be caused by the failure in spike propagation along collateral branches (Luscher, 1990). For a simple case the involvement of this factor is described by the RV:
D[r/] = Q2(O[~C] "E[O] +O[O] "E2[sc]) A[r/]=Q3(A[~].E[O]+E3[~
+ 3 E [ sc ] • D[~:]. D [ O ] ) .
o
r / = Q ~] ~:i,
c].A[O] (25)
Analytical investigation of expressions for R 1 and R 2 proves that these values are situated under binomial RPV, if sc and O are binomial RVs.
(24)
i-1
where O is RV modeling the number of contact zones at which spike has been arrived. Suppose that transmitter release in the zones follows the same rule (~:). In order to obtain moments of RV
Modeling of non-linear summation Taking into consideration the non-linearity between amplitude of PSP and conductance (Martin,
-5700 I
B
-5720 "~
-5740
O ,t= •-
-5760
\
o
-5780 O
-5800 50
i
r
i
70
90
110
\
\
i
130
150
Quantal s i z e (j~V) 50
/
40
8-1
20 10 0 -0.05
0.5 EPSP amplitude (mV)
Fig. 3. A: time courses of smoothed individual and averaged sensorimotor EPSPs. B: dependence of logarithm of the likelihood function on the quantal size. Optimal value of the size (Q) is 70/xV. C: averaged EPSPs with amplitude in the range Q +_ 1 / 2 • Q, 2 • Q _+ 1 / 2 • Q , . . . , 7- Q _+ 1 / 2 • Q. D: histogram of amplitude and fitted density of unconstrained quantal model.
209 1.00
1977), the revealed potential v is related to the number of released quanta (k) by
v
=
kQ . F / ( k Q
+
F),
0.80 0.60 0.40
(26)
where F is the driving force. Supposing that the number of released quanta is described by binomial RV, negligible increase of R 2 over simple binomial model RPV has been found (Fig. 2C).
•
0.20
A
0.00
•
.-'-~"° • //
/~
1/
•
2
/
-0.20
l////~f
-0.40
/I/
-0.60
.
-0.80
,
////
,
,
i
i
i
i
e
- 1.00
1.00
Results
0.80
R2
0.60
Comparison of experimental and theoretical values of R 1 and R 2 Twenty-one Ia a f f e r e n t - m o t o n e u r o n connections were taken for statistical analysis. Individual EPSPs and steps of evaluation of the quantal size are shown in Fig. 3. Only for 3 amplitude samples, the values of R 1 and R 2 were situated lower than RPV for the binomial model (Fig. 4A). If EPSP fluctuation were described by binomial model, then about half of the points would fall below the line (Fig. 4B). This discrepancy is significant ( P < 0.01; chi-square test). The rest 18 points lay inside the binomial convolution RPV. Most of the experimental points form a cluster with the center inside beta-model RPV (Fig. 4A). The cluster is farther from RPV for Myamoto's model than from the beta-model region. It seems plausible that the scatter diagram observed is a result of statistical fluctuation of R 1 and R2 values. To verify this suggestion we simulated 20 samples of PSP amplitudes described by betamodel with values of parameters a = 0.17 and b = 0.68. These values were obtained from the expressions 10 and 11, where RI and R 2 were the coordinates of cluster center. The scattergram obtained after simulation (Fig. 4C) turned out to be similar to the experimental one. This leads to a hypothesis that Pi distributions at the connections under study can be described by a common law. Regions of permissible values of models incorporating decremental spreading, lateral interaction and occlusion are presented in Fig. 5. The sets of curves were drawn for the connection shown in Fig. 3 (Grantyn et al., 1984b). Evidently, only the model with lateral interaction gives val-
0.40
~t
0.20
B
0.00 g•
-0.20
•
-0.40 -0.60 -0.80 - 1.00
1.00 0.80 0.60 0.40 O2.0 C
0.00
0.20
, t...~
-0.40
-0.60
-o.ao -
1.00
0.00
. .~
.-/
•
/
R, i
0.20
~
040
o.eo
o.ao
e
1.00
Fig. 4. Scatter diagram of values R s and R 2 calculated for 21 samples of frog sensorimotor EPSP (A) and simulated values of R 1 and R 2 for binomial model (B) and beta-model (C). + is the center of points. Parameters: binomial model, p = 0.6, n = 15; beta-model, a = 0.17, b = 0.68, N = 45. Sample size was 400.
ues of R 1 and R 2 situated above the RPV for binomial model. This is a case in which the values of parameter r are negative, i.e., under lateral inhibition. However, even then the experimental point is situated above RPV. Shapes of the curves for 4 other connections were very similar to the above and the revealed relationships between binomial RPV and values of R 1 and RE calculated for these three models were presented.
210 TABLE 1 ESTIMATES OF T H E PARAMETERS OF T H R E E Q U A N T A L RELEASE MODELS A N D M O R P H O L O G I C A L PARAMETERS OF F R O G S E N S O R I M O T O R CONNECTIONS No.
Quantal size
Noise SD
Binomial model
Binomial convolution model
Betamodel
(/..tV)
(#V)
n
p
NI
Pt
N2
Nt + N2
N
M0
Number Bouton
Contact zone
40 43 42 72 26
11 14 15 23 8
1 2 3 4 5
67 57 70 92 60
30 38 29 48 30
16 22 16 20 9
0.63 0.38 0.66 0.72 0.60
17 29 14 18 9
(I.29 0.26 0.44 0.36 0.49
5 1 4 8 1
22 30 18 26 10
50 43 52 74 26
6 3 4 9 2
6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
109 73 71 71 79 61 57 122 80 91 61 61 91 76 81 62
43 27 33 77 41 36 57 52 35 40 56 77 36 37 28 32
17 17 15 37 6 14 16 11 22 13 16 8 3 13 7 12
0.81 0.71 0.70 0.39 0.63 0.47 0.54 0.60 0.40 0.62 0.28 0.66 0.69 0.61 0.58 0.51
10 17 15 . 7 15 16 11 32 13 . 7 3 16 7 13
0.54 0.25 0.27 . . 0.35 0.39 0.40 0.62 0.20 0.55 . . 0.58 0.69 0.33 0.44 0.31
8 8 6 . 1 1 2 0 2 1 . 1 0 3 1 2
18 25 21
10 8 6 8 I 1 2 3 4 3 1 1 0 4 1 2
Have not measured
8 3 19 8 15
69 62 51 74 17 34 43 33 43 42 23 25 11 41 20 31
42 14.6
13 5.1
0.60 0.09
14 6.9
0.41 0.13
17 7.9
40 15.1
3.7 2.4
45 16.8
models
are
Mean SD
76 17.3
Estimates o f quantal release parameters Evaluated
parameters
given in T a b l e
I. T h e
of
mean
three number
8 16 18 11 34 14
2.9 2.4
units of quantal
times
predicted the
by beta-model
binomial
(N)
parameter
differ from the number
14 5.6
n
of boutons
exceeded and (t
did
3 not
test, P>
TABLE II ESTIMATION OF C O N T R I B U T I O N OF SEVERAL FACTORS BY Q U A N T A L ANALYSIS OF THE F L U C T U A T I O N OF PSPs Factor
Skewness test a
Morphological test b
Spatial non-uniformity Pi Temporal non-uniformity p Variation of quantal size Decrement spreading of local PSP Lateral inhibition Occlusion Non-linear summation Failure
+4-
+
+ + + + -
+ + + -
~' Two crosses: agreement between theoretical and experimental values of variation and skewness indexes; one cross: values of skewness index are more than binomial one; minus: reverse relationship. b Cross: the number of quantal units predicted by binomial model (n) is smaller than real value; minus: reverse relationship.
211
0.1). Correlation coefficient between the number of boutons and N was 0.95 (5 connections). Binomial parameter n was close to the numbers of contact zones (correlation coefficient cor. = 0.84) and to the number of units functioning with probabilities Pi > 0.2 (cor. = 0.85). The latter number was predicted by beta-model. The estimate of mean probability of quantum release obtained by beta-model was approximately 0.2. The mean number of the quantal units estimated by binomial convolution model ( N 1 + N 2) was more than binomial n (17 vs. 13), being however much lower than that of boutons (45).
A
x
=
2
S
0
a 0 0.00
0.20
i
I
h
0.40
0.60
0.80
1.00
Release probability
1000;,500;300;~
The number of high-effective units predicted by binomial convolution ( N 2) were close to the minimal values of EPSP amplitudes divided by quantal size (M0). Fig. 6 represents shape of the distribution of release probabilities estimated by the beta-model. This curve corresponds to mean values of R~ and R 2 (cross in Fig. 4A and C) and gives integrative representation of the non-uniformity of the release probabilities at frog sensorimotor synapse. The density has two maxima at 0 and 1, the first being much expressive. The shape of the density curve agrees very well with the fact that the estimated number of high-effective units ( N 2) is much smaller than the total number of quantal units. The value of N 2 corresponds to the number o f quantal u n i t s w i t h Pi > 0.8 ( c o l = 0.8, 5 experiments).
- 1.00
i
=
. , .;.;;
.
O.6O
0.40 ÷ O.2O
B
I
Fig. 6. Estimated density of release probability in single boutons of frog sensorimotor connection (a = 0.17, b = 0.68).
tOO p~ 0.80 0.60 0.40 0.20 0.00 -0.20 -0.40 -0.60 -0.80
tOO 0.80
8
0.00 -0.20 -0.40 -0.60 -0.80 - 1.00
tOO c
0.80
C
=
0
.
3
~
0.60 0.40 0.20 0.00
Discussion
-020
--0.40 --0.60 --0.80 - 1.0,£
0.00
0.20
040
I
L
I
0.60
0.80
1.OO
Fig. 5. R 1 - R z relationships for several values of the electrotonic length ,~ (A), the interaction coefficient r (B) and the occlusion coefficient c (C). + is the value of R s and R 2 calculated for experimental sample of EPSP amplitudes.
Overestimation of the quantal size obtained by the maximum likelihood method (deconvolution procedure) as a parameter of model of mixture of normal distributions has been reported (Clamann et al., 1991). In the present study the unconstrained quantal model was used. The number of parameters to be estimated was nearly 2 times less, compared to the former model. Moreover, no correlation of quantal size and noise SD was
212
found (cor. = -0.011). This is in contrast to the results obtained under the study of synaptic transmission at cat interneuronal connections by several workers (Clamman et al., 1991). Moreover, if true values of quanta were less than those used for calculation of R 1 and R2, true points would be situated along the line connecting estimated values with the coordinate center and farther on the center than shown on Fig. 4A. Such displacement, however, would not be essential for the results presented here. It may be possible that a stimulating electrode activated a few rather than one afferent fibers. This could be the cause of high values of the asymmetry coefficient R 2 (Fig. 2B). However, this is unlikely because the amplitude of electrical component of frog single-fiber sensorimotor EPSP did not fluctuate (Grantyn et al., 1984b). The mean value of R 2 calculated under extra- and intracellular stimulations did not different statistically. Table II summarizes the results obtained in the study of the influence of several factors on the values of skewness index and on the number of quantal units predicted by binomial model. We conclude that only involving of spatial non-uniformity it is possible to account for the discrepancies of binomial and experimental values. On the other hand, it is not to be excluded that lateral inhibition of neighboring contacts and variation of the quantal size also contribute to fluctuation of frog sensorimotor EPSPs. The model of the convolution of two binomial distributions was sufficient to describe the scattergram of experimental values of R~ and R 2, but the predicted release sites were much less numerous than the observed. In contrast to this model, the beta-model gives a good explanation of the obtained scattergram and accurately predicts the number of contacts. Addition to the beta-model of the pool of contacts in which each impulse induces transmitter release gives RPV with high values of R 2. It can be expected for interneuronal connection of cat where high-effective contacts were revealed (Walmsley et al., 1988). Beta-model gives accurate interpretation of binomial n as the number of contacts in which quantal release occurs with probability of more than 0.2. Hence, considerable
underestimation of the number of contacts by binomial n is due to involvement in synaptic transmission of a large number of low-effective (Pi < 0.2), contacts which are responsible for the peak at 0 on Fig. 6. List of abbreviations
EPSP PSP RPV A a
B(a, b) b c D d E F G
M3 N Nl, N2
Pl, P2
ff p
Pi Q Qi Ri R2
S2
excitatory post-synaptic potential post-synaptic potential region of permissible values third central moment p a r a m e t e r of beta-distribution beta-function parameter of beta-distribution p a r a m e t e r defining extent of occlusion variance, dispersion part of quantal units functioning with
pi = 1 expectation driving force number of contact zones estimated third moment of the number of quanta released total number of non-uniform quantal units numbers of less and more effective quantal units binomial parameter, the total number of uniform units release probabilities of less and more effective units probability of single-fiber activation binomial parameter, release probability of uniform units mean release probability release probability of ith quantal unit quantal size contribution of ith quantal unit to recorded potential ratio between the variance and mean number of quanta released ratio between the third moment and variance of quanta released degree of interaction of adjacent contacts estimated variance of the number of quanta released
213
~7
mean number of quanta released length constant number of quanta released amplitude of recovered distribution of PSP
Acknowledgements
We would like to thank Dr. N.M. Chmykhova for morphological data, Prof. P. Clamann for critical reading and helpful comment on the manuscript, I.I. Evdokimov and A.B. Veremiev for programming assistance.
References Babalian, A.L. and Chmykhova, N.M. (1987) Morphophysiological description of connections between single ventrolateral tract fibers and individual motoneurones in frog spinal cord, Brain Res., 407: 394-397. Bart, A.G., Dityatev, A.E. and Kozhanov, V.M. (1988a) Quantal analysis of the postsynaptic potentials in the interneuronal synapses: recovery of a signal from the noise, Neirofiziologiia (Kiev), 20: 479-487. Bart, A.G., Dityatev, A.E. and Kozhanov, V.M. (1988b) Analysis of the transmission in the interneuronal synapses by the convolution of binomial distributions, Neirofiziologiia (Kiev), 20: 487-494. Bennett, M.R. and Lavidis, N.A. (1979) The effect of calcium ions on the secretion of quanta evoked by impulse at nerve terminal release sites, J. Gen. Physiol., 74: 429-456. Chmykhova, N.M., Shapovalov, A.I., Shiraev, B.I., Komissarchik, J.J. and Snigirevskaja E.S. (1987) Ultrastructural analysis of physiologically identified and stained with horseradish peroxidase the sensorimotor synapse of frog, Doklady AN SSSR (Physiol.), 293: 249-252. Clamann, H.P., Rioult-Pedotti, M.S. and Luscher, H.R. (1991) The influence of noise on quantal EPSP size obtained by deconvolution in spinal motoneurones in the cat, J. Neurophysiol., 65: 67-75. Del Castillo, J. and Katz, B. (1954) Quantal components of the end-plate potential, J. Physiol., 124: 560-573. Dityatev, A.E. (1991) Analysis of non-uniformity of transmitter release probabilities in frog interneuronal synapses. Ph.D. Thesis, Sechenov Institute of Evolutionary Physiology and Biochemistry, 186 pp. Dityatev, A.E., Gapanovich, S.O. and Kozhanov, V.M. Influence of extracellular Ca 2+ on transmitter release parameters at Ia-afferent fiber connections on motoneurones of frog (submitted to Neuroscience).
Edwards, F.R., Redman, S.J. and Walmsley, B. (1976a) Statistical fluctuation in charge transfer at Ia synapses on spinal motoneurones, J. Physiol., 259: 665-688. Edwards, F.P., Redman, S.J. and Walmsley, B. (1976b) Nonquantal fluctuations and transmission failures in change transfer at Ia synapses at spinal motoneurones, J. Physiol., 259: 689-714. Faber, D.S., Funch, P.G. and Korn, H. (1985) Evidence that receptors mediating central synaptic potentials extend beyond the postsynaptic density, Proc. Natl.Acad. Sci. USA, 82: 3504-3508. Feller, W. (1984) An introduction to probability theory and its applications, 3rd edn., Vol. I, John Wiley, New York, 527 PP. Grantyn, R., Shapovalov, A.I. and Shiriaev, B.I. (1984a) Tracing of frog sensory-motor synapses by intracellular injection of horseradish peroxidase, J. Physiol., 349: 441-458. Grantyn, R., Shapovalov, A.I. and Shiriaev, B.I. (1984b) Relation between structural and release parameters at the frog sensory-motor synapses, J. Physiol., 349: 459-474. Hatt, H. and Smith, D.O. (1970) Non-uniform probabilities of quantal release at the crayfish neuromuscular junction, J. Physiol., 259: 395-404. Katz, B. and Miledi, K. (1963) A study of spontaneous miniature potentials in spinal motoneurones, J. Physiol., 168: 389-422. Korn, H. and Mallet, A. (1984) Transformation of binomial input by the postsynaptic membrane at a central synapse, Science, 225: 1157-1159. Korn, H. and Faber, D.S. (1990) Transmission at a central inhibitory synapse. IX. Quantal structure of synaptic noise, J. Neurophysiol., 63: 198-222. Korn, H., Mallet, A., Triller, A. and Faber, D.S. (1982) Transmission at a central inhibitory synapses. II. Quantal description of release, with a physical correlate for binomial n, J. Neurophysiol., 48: 679-707. Kuno, M. (1964) Quantal components of excitatory synaptic potentials in spinal motoneurones, J. Physiol., 175: 81-99. Kuno, M. and Weakly, J.N. (1972) Quantal components of the inhibitory synaptic potential in spinal motoneurones of the cat, J. Physiol., 224: 287-303. Luscher, H.R. (1990) Transmission failure and its relief in the spinal monosynaptic reflex arc. In: M.D. Binder and L.M. Mendell (Eds.), The Segmental Motor System, Oxford University Press, New York, pp. 328-348. Martin, A.R. (1977) Junctional transmission. II. Presynaptic mechanisms. In: E.R. Kandel (Ed.), Handbook of Physiology, Sect. 1, Vol. 1, Pt. 1, Baltimore, MO, pp. 329-355. Miyamoto, M.D. (1986) Probability of quantal transmitter release from nerve terminals: theoretical consideration in the determination of spatial variation, J. Theor. Biol., 123: 298-304. Neale, E.A., Nelson, P.G. and McDonald, R.L. (1983) Synaptic interactions between mammalian central neurons in cell culture. III. Morphophysiological correlates of quantal synaptic transmission, J. Neurophysiol., 49: 1459-1468. Pun, R.Y.K., Neale, E.A., Guthrie, P.B. and Nelson, P.G.
214 (1986) Active and inactive central synapses in cell culture, J. Neurophysiol., 56: 1242-1256. Redman, S. (1990) Quantal analysis of synaptic potentials in neurons of central nervous system. Physiol. Rev., 70: 165198. Shapovalov, A.I., Shiriaev, B.I. (1987) Signals Transfer in Interneuronal Synapses, Nauka, Leningrad, 173 pp. Tuckwell, H.C. (1988) Introduction to theoretical neurobiology. Vol. 1. Linear cable theory and dendritic structure, Cambridge University Press, Cambridge, 291 pp. Voronin, L.L., Kuhnt, U. and Gusev, A.G. (1991) Analysis of EPSP fluctuations indicated increased presynaptic release during long-term potentiation in area CA1 of hippocampal slices, Neurosci. Res. Comm., 8: 87-94. Walmsley, B., Edwards, F.P. and Tracey, D.J. (1988) Nonuniform release probabilities underlie quantal synaptic trans-
mission at a mammalian excitatory central synapse, J. Neurophysiol., 60: 889-908. Wong, K. and Redman, S.J. (1980) The recovery of a random variable from a noisy record with application to the study of fluctuations in synaptic potentials, J. Neurosci. Methods, 2: 389-409. Yamamoto, C., Higashima, M., Sawada, S. and Kamiya, H. (1991) Quantal components of the synaptic potential induced in hippocampal neurons by activation of granule cells, and the effect of 2-amino-4-phosphonobutiric acid, Hippocampus, 1: 93-106. Zefirov, A.L. and Stolov, E.L. (1982) Model of transmitter secretion in neuromuscular synapse based on the spatial non-uniformity of release probability of acetilholin quanta, Neirofiziologiia (Kiev) 14: 233-240.
