The cerebellar cortex compartment model and the characteristics of (mossy fibres)-(granule-cell) activity transformations were described in a preceding article.
Biological Cybernetics
Biol. Cybern. 51, 407415 (1985)
9 Springer-Verlag 1985
Computer Simulation of a CerebeHar Cortex Compartment II. An Information Learning and Its Recall in the Marr's Memory Unit W. L. Dunin-Barkowski and N. P. Larionova Institute for Problems of Information Transmission, USSR Academy of Sciences, Moscow, USSR
Abstract. Computer simulation experiments are described regarding information storage and retrieval at a network consisting of one Purkinje cell and 20,000 granule cells. The information content depends on a scheme type and the properties of Purkinje cells. It is shown that a practically attainable information record efficiency is of the order 0.6 bit per binary memorising synapse. Associative information recall is demonstrated for the Marr's memory unit and expressions are derived for an information-content estimation based on parameter values obtained by simulation. The consequences of this computer simulation for physiological experiments are extensively discussed. The cerebellar cortex compartment model and the characteristics of (mossy fibres)-(granule-cell) activity transformations were described in a preceding article (Dunin-Barkowski and Larionova, 1985). The main scheme parameters were justified in that article with the aid of a simple theory (Dunin-Barkowski, 1978). The computer simulation experiments regarding information learining by a Purkinje cell and information recall are described in the present article. Definitions and the notation used here are the same as in (DuninBarkowski and Larionova, 1985). We obtained certain parameter correlations which must satisfy the cerebellar cortex circuitry if Marr's hypothesis is to truly describe the main functions of the cerebellar cortex. Besides, we give a list of "experimental information which is absent", but necessary in any attempt to detalise the theory of the cerebellar cortex.
1 Information Learning and Its Recall with Complete Events There were two distinct procedures used in the simulation: (1) the learning of events and (2) the recall of stored information.
The procedure regarding the learning of events consisted of a random choice of n different events ( l < n < 7 0 ) of the size Lo=357 in a set of No=700 mossy fibres and of the modifications of those synapses on a Purkinje cell whose granule cells participate in any chosen event. During the recall of stored information, a set of randomly chosen events was delivered to the mossy fibres. The number of excited granule cells and a number of excited cells with modified synapses were calculated for each event. After this procedure, the information content, which may be extracted from the scheme, was evaluated in accordance with the laws of Purkinje-cell excitation. For the letter, the weights of the modified synapses as well as the Purkinje cell threshold were essential. Though " + "-learning and "-"-learning imply highly different Purkinje cell excitation laws, there exists an accordance of parameters which makes these cases equivalent with regard to the quantity of information stored. Owing to this fact, we consider (here) only the case of "+"-learning. Various consequences of these two possibilities will be dealt with in No. 3. We assume am, a, and h to be the weights of the granule-cell synapses (in modifyed and unmodifyed states) and the Purkinje cell threshold, respectively, and consider the following cases: 1. a, = 0; am = 1; h is chosen to make the amount of the information obtained after the recall procedure (see lower) a maximum. 2. a , = - 1 ; a,,=l/N;h=l/N. 3. The synaptic weights are chosen according to 2., but the threshold value is established in order to maximize the amount of the information retrievable from scheme. The Purkinje cell is excited or inhibited by a certain input event, depending on its excitation law. The number of all possible events, which may appear in
408
the mossy-fibre input, is very large: No) L0
=
2No.S(Lo/No)'
i.e. 2 T~176 for our conditions. Due to this fact, we can not test all possible events and must evaluate the PCexcitation probability during simulation for the two opposite conditions: when the input event is learned or unlearned. In principle, this probability can be evaluated empiricaly, i.e. we must give a sufficient number of events to the scheme in order to obtain an empirical estimate of the probabilities. This method of probability evaluation demands that a lot of events be tested and is practically impossible for most parameter ranges. We restricted the test number by the requirement to have the necessary data to evaluate the excited and learned granule-cell number distribution. The desired probabilities were calculated on the assumption that the distributions obey the normal law for empirical values of the mean value and the dispersion. The obtained probabilities (we call them the falsememory and missing probabilities) were used to calculate the information capacity of the scheme. The formulas are given in the Appendix. Figure 1 gives an example of the learned and unlearned excited granule-cell number-distribution histogramms when unlearned events are at mossy-fibre inputs. It is evident that the distribution for learned events coincides with the unconditioned excited granule-cell number distribution. Figure 2 represents the PC-information content as a function of a PC threshold at two values of the learntevent number, n, and two different values of the excited granule-cell number distribution dispersion, a (only for n = 17). The data are given separately (Fig. 2A and B) for the two values of the an/a m ratio ( - N and 0, respectively). The I(h) function has a maximum at the optimal-threshold value. This maximum is sharply expressed at a,/am = 0 and smooth at o-,/o-m= - N . In the latter case, the PC-information content at a nearzero value of h is relatively high, though the ! value is significantly larger at optimal h than at h-~ 0.
o
s
qO0
6o0
loo
Fig. la--e. The excited granule-cell number distribution histogramms of the learned (a), unlearned (h), and all synapses (e) for 100 randomly chosen input events. The B-scheme. The 12 events are learnt. P is the portion of all events in a given abscissa interval of the lengh 10
50001I
A .
.
.
.
.
.
.
.
~o000
5000
I 7000
200
qO0
GO0
~00
B
5000
5000
000
h. 2,00
qO0
6OO
~00
Fig. 2A and B. The plot of the PC information content vs. the PC threshold. The B-scheme. A a J a m = - N ; 1 n=8, a=25; 2 n = 1 7 , a=25; 3 n=17, a=20. B a Jam=0; 1 n=8, a=14; 2 n=17, a=16; 3 n=17, a=19. The value o f / i n bits
Figure 3 represents all the results obtained from different simulation experiments on the evaluation of an information-content dependence on the learntevent number. For the sake of comparison, a theoretical curve is given which was obtained according to (Dunin-Barkowski, 1978). We feel that the following data pecularities are mostly substantial. 1. The maximum information content of the schemes with large excited granule-cell number dispersions (D-, DR-, SC-schemes) is very limited (see Fig. 3). Possibly, the reason lies not in the big dispersion, itself, but is due to the tight dependence of the different granule-cell excitations. Thus, the Purkinje cell does not have N independent synapses, but substantially less. This result indicates that the properties of a mossy fibre with regard to a granule-cell connection scheme are crucial for the learning cerebellar mechanism postulated by Marr. 2. The highest efficiency is for the case when the PC threshold is optimal and cr,/o-,,=-N for the B- or B 143-schemes. In this case, the information efficiency is of the order 0.6 bit per binary memorizing synapse: this value is in a close vicinity to ln2=0.69 the value
409
T
7OO0,
6 /
40
50
$0
/
70
Fig. 3a and b. The PC information content (in bits)vs, the learnt event n u m b e r for the different connection types between M F s and GrCs and some PC exitation laws. a The B-scheme; 1 a,flr, = 0, optimal h; 2 g , / g , = - N, h = 0; 3 same as in 2, but at optimal h. b (in the dashed circle): cr,/~,=0, 4 the D-scheme, 5 the DR-scheme. T is the theoretical curve (a) (DuninBarkowski, 1978; Dunin-Barkowski and Larionova, 1985)
which constitutes the upper limit :of the efficiency of Marr's memory unit (Dunin-Barkowski, 1978). This means that the cerebellar scheme (with reasonable connections) may have such a high efficiency that it is theoretically possible. 3. It should be noted that in the B- and B143schemes, the connection randomness is substantial. Meanwhile, in a process of the random-connection choice there is a danger of obtaining a scheme with zero efficiency. The "bad scheme" probability is small when the scheme is large enough (Dunin-Barkowski, 1978). Our computer-simulation results indicate that N = 20,000 is a large enough number to provide a nice scheme. 2 Associative Information Reproduction
It is easy to prove that the amount of information stored does not depend on the subevent size if the scheme can differentiate all the subevents of the learnt events from events of the same size which are not the subevents of the learnt events. This proposition is true for a subevent size equal or exceeding 2 and in the absence of errors in the scheme work. The simple theory of (Dunin-Barkowski, 1978) can not be applied to the associative recall. We explore this phenomenon in our simulation model (only for the B-scheme type). We store in the scheme n events of the size Lo = 357, as in the preceeding section. The information recall
consisted of a presentation at mossy-fibre inputs of the subevents of the learnt event and of random events of the appropriate size (we call them random subevents). During a presentation of events (with reduced sizes in comparison with the learnt event) to mossy fibres, we deminish the granule-cell threshold to imitate the Golgi-cell function. Three main Golgi-cell activation rules were considered: (a) a complete stabilization of the active granule-cell number; (b) a linear dependence of the inhibition on an active granule-cell number; (c) a constant granule-cell threshold (the absence of a Golgi cell). For full-sized events (Lo = 357) all these rules give identical threshold values. The active granule-cell number and the granule-cell threshold dependence on the active mossy-fibre number for different Golgi-cell activation rules are plotted in Fig. 4. We do not consider the dynamics of the granulecell threshold setting. In such circumstances, there is no difference in how the Golgi-cell activation is achieved: by the mossy-fibre firing or by the granule-cell firing. The granule-cell output is determind by the granulecell threshold dependence on an excited mossy-fibre number. This dependence is step-like because the values of a threshold may be only integer. Due to the fact that all the granule cells are identical, the step-like threshold curve invokes the saw-like dependenceof the excited granule-cell number on the excited mossy-fibre
E
A
4 4 3 //
2,
t,
Lo ,L
500,
B 5
/"i
3001 oo,
_.
,
&
zbo Fig. 4A and B. Different Golgi-cell control modes. A The G r C threshold vs. the active M F s number plot. 1 g = const; 2 ~ = 1 ยง 4 9(L/690); 3 such a control rule that keeps constant the active GrCs number. B Active GrCs number vs. active M F s number. 1 without control; 2 linear threshold control; 3 stabilising control. Dashed lines are smooth envelops of the curves
410
number (Fig. 4). This is in accordance to the Szentagothai and Pellionisz data (Pellionisz and Szentagothai, 1973). An introduction of the inhomogenity of granule cells can make these dependencies smooth (just like the dashed lines in Fig. 4) (Torioka and Koga, 1977). As was mentioned above, we tested three ways for granule-cell activity control: (1) no control, GrC threshold is 5; (1) linear control, GrC threshold is given by the formula n = 1 +L/690, where n is the GrC threshold, L is a number of active granule cells; (3) stabilizing control which provides the active-granulecell number being constant at Lo = 5, 44, 116, 216, and 357. Different active-cell number control ways show different associative efficiences, as is described below. No control, i.e. a constant granule-cell threshold, implies that the active-granule-cell number rapidly decreases (in accordance to the law x~,x~)) with a decrease in the active mossy-fibre number. In this case, the false-memory probability increases, as follows. If the active-granule-cell number, excited by the present mossy-fibre input, is L, then the false-memory probability is pL (p is the learned synapses fraction). When L is small pL is relatively large. The granule-cell threshold control allows the excitation of a sufficiently large granule-cell number. This implies, however, a recruitment of not only true but also false granule cells, which are not active when the input events are learnt. Probably, there exists an optimal threshold control, which establishes the "best" ratio between true and false excited granule cells. The excited learned granule-cell number distribution histograms, obtained by the presentation to the mossy-fibre input of learnt and unlearnt subevents, are presented in Fig. 5.
1I
It is evident that in every instance these histograms are different. This means that the investigated schemes possess associative abilities. However, a sure distinction between a learnt subevent and random "subevents" is possible only in the case of relatively large-sized subevents. It is interesting to evaluate the information content extractable from learnt schemes. For this purpose we use the expressions which are given in the Appendix. The Fig. 6 includes plots of information content versus information presented for learning with different sizes of subevents. Two points are important in information-content calculations. The first is that the expressions mentioned above are obtained by a relatively rough approximation (see Appendix). The other is due to the fact that the estimates are made only for a PC excitation mode with a, = 0, a,, = 1 and the optimal threshold. Meanwhile, this mode is not highly effective, even when the full events are used for the information extraction (see Fig. 3). In spite of the above mentioned deficiency, the information-content evaluation clearly shows the following features: (1) the Marr's memory unit possesses associative properties; (2)these properties depend on Golgi-cell firing laws: (3) the associative properties vanish when the size of a subevent becomes small. The following pecularity of the associativeinformation recall in our schemes should be emphasized. The Lo-sized event number on No mossy fibres is ( NL0~
(
No~. In order to extract the information stored in the L0J scheme, we must present (for mossy fibres) either all events of the size L 0 or all "subevents" of the size E0.
.I
f.2t-.--i
Z
d_L 0
I 0
gO0
~ qO0
3
/0
m 600
0
500
500
The number of Eo-sized subevents is
700
Fig. 5a-h. The active and learned granule-cell number distribution histogramms when learnt subevent and "random subevents" are given to the scheme. I For learnt subevents, I I for unlearnt subevents, 1II the whole excited GrCs histogramms. a--e Linear threshold control; d no control; e--h stabilising control. Lo values are: a 20, b 90, e 200, d 116, e 5, f44, g 116, h 216. Some 17 fttll events are learnt. The GrC threshold is indicated at the right side of each line. The vertical bar (denoted 20 in the middle) corresponds to 20 counts per abscissa interval of the length 10 (a--e and e--h) and of unit length in d
411
~000 350(
3001 250(
II
200(
t50~ I
~000
I ,
o-5 ,-'i
,,
500 li~ i
x-~ A-2 m-,i 6
8
~z
zo
"
Fig.6.The informationcontent(inbits)ofthe B-schemevs.the learnt-event number for the associative information recall. The G r C threshold is as indicated at each curve. The PC threshold during recall is optimal, a. = 0; am = I
The latter number is much less than the former if L o < ( 1 / 2 ) . N o and E 0 < L o. A value No
No
The physiological meaning of (1) depends on the type of PC-learning ("+ "-learning, or "-"-learning). Let a+ be the excitatory synaptic weight of an acting granule ceU-Purkinje cell synapse and o-_ be the amount of the final inhibitory influence of an active granule cell on a Purkinje cell through the cortical inhibitory interneurons. It should be kept in mind that a number of granule cells, which have an inhibitory influence on the given Purkinje cell, differs from N, i.e. the number of granule-cell synapses on a Purkinje cell. This value N~ is 10-20 times larger than N [data for the cat (Smolyaninov, 1971)]. It should be noted, that according to postulates adopted in (Dunin-Barkowski and Larionova, 1985, p. 403) the probability of GrC firing is the same as for the set of all GrCs which directly synapse on a given PC and for the set of all GrCs which inhibit the PC through the I n l s . Thus, if L granule cells (directly acting on a given PC) are excited it is almost certain that ( L / N ) . N ~ = L . ( N I / N ) granule cells which inhibit this PC are excited. In other words, the activity of one granule cell, which acts directly on a Purkinje cell, corresponds to an activation of N I / N granule cells acting on this cell through inhibitory synapses. We suppose that in the "+"-learning model the weight of an unlearned granule-cell-Purkinje cell synapse is zero and the weight of the learned synapse is a + (and vice versa for the "-"-learning model). The weight of the inhibitory influence of a granule cell on the Purkinje cell is unmodifiable by learning and constitutes, as mentioned above, N ~ / N . a _ . These suppositions give the following expressions for the values of the learnt and unlearnt influences on the Purkinje cell for " + " - and "-"-learning models a n = -- N f f N .
a_,
a m= - NI/N.
a_ + a +,
may be called the "information gain" of the associations. It is easy to see that the testing of the learnt scheme using subevents provides a b-fold increase in the extractable quantity of information per memory test, when compared to testing using full events.
for the "+"-learning model and
3 Simulation-Based Predictions of Experimentally Measurable Parameters
for the "-"-learning model. Taking (1)-(3) as the basis for obtaining of expressions for a + and a_, which depend on the type of learning, we have
Addressing the scheme by full events we show that the largest values of the scheme information content are obtained at a , / a m .~ - N .
(1)
In a real cerebellum, values o-. and a mare provided by the parallel fibre-Purkinje cell synapses and by the disynaptic path: the granule cell-cortical inhibitory interneuron-Purkinje cell.
an=
-
NI/N.
a,, = - N l / n .
a_
+ a
(2)
+,
(3)
a_ ,
a +/a_ = (NI/N). (N + 1)IN ~ N,/N,
(4)
for the "+"-learning and a+/a_ = (NJN).
N +
1) ~ NI,
(5)
for the "-"-learning. The ratio a + / a _ in the latter case is N times greater than that for the preceeding one.
412
Thus, it may be concluded that if the cerebellar interconnections are arranged so as to provide (1), the ratio a+/a_ ought to follow (4) or (5), depending on the learning type. It is evident that both the learning type and the ratio o-+/a_ can be experimentally determined by independent measurements. From this fact [whether such experiments would satisfy (4), (5) or not] we can know the consistency of our learning theory. The other experimental expectation is based on the fact that schemes possessing a relatively large activegranule-cell number dispersion (cf. Fig. 3) are not effective for information storage. This fact implies that the active-granule-cell number dispersion should be relatively small (of the order of Poisson's distribution) if the cerebellum is effectively used for information storage. The experiment proposed in (Dunin-Barkowski, 1978) should to be described here as follows. Pairs of the Purkinje cells with a common climbing fibre must be recorded. We can make sure that two PCs have a common climbing fibre if their complex spikes are synchronous. If Marr's theory is correct, these Purkinje cells will have a high correlation of the simple spike activity. This experiment is crucial: if different Purkinje cells with a common climbing fibre have different clear functional correlates [such as in (Noda and Suzuki, 1979, Fig. 2)] of their simple spike activity, then Marr's theory is not applicable to cerebellar structures. 4 What is Urgent but Still Unavailable for the Creation of an Effective Cerebellar Model? One of our goals is to elucidate which data are vitally urgent for an elaboration of effective cerebellar models. In other words, one of the simulation goals is not to predict experimentally measurable values, but to point out which questions would be answered if some definite data were to be obtained.
neurons, the Purkinje cells, the inhibitory cortical interneurons etc., is necessary. It is also necessary to have information on the output-spike generating mechanism in all cerebellar neurons. Accurate data are necessary for synaptic modification.
4.3 The Characteristics of the Net Activity Methods are needed to evaluate the individual impulse frequences in all cortical neurons and the statistical characterization of these frequencies. These data would provide some knowledge regarding the connection geometry (Dunin-Barkowski and Larionova, 1985). When the latter is known, the same data may be used for checking the hypothesis regarding the spike-generating mechanism of individual neurons.
4.4 The Overall Cerebellar Performance Methods and techniques are needed for the qualitative and quantitative characterization of information processing and storage amounts which are necessary for the tasks performed by the cerebellum. These data would enable us to compare the results from a calculation of informational parameters values, which can be implemented in neuronal structures of the cerebellum, with macroscopic data. It also would allow us to predict the values of measurable parameters for new experiments. Appendix
Evaluation of Information Content of the Mart's Memory Unit on the Basis of Computer Simulation Data 1
4.2 The Physiology of Neurons
1 The Values of Pfm and Pmiss" These values can be calculated on the basis of definite Purkinje cell excitation rules. The value Pfm is the false-memory probability. It defines the risk of a Purkinje-cell excitation when a non-recorded event arrives at mossy fibres. The value Pmissis the probability of the non-excitation of a Purkinje cell in response to a learnt event. Let F(L) be the excited granule-cell number distribution. The values Fm/~(L)and Fm/s(L) are the excited and learnt granule-cell number distribution for learnt and unlearnt events respectively. The values F./t(L) and F./~(L) are the corresponding distributions for unmodified synapses. An estimation of Pfm and Pmlss needs, in general, a knowledge of distributions of the weighted sums of learnt (weight am) and unlearnt (weight a,) synaptic actions of excited granule cells when learnt and unlearnt events are at the inputs.
A quantitative characterization of an interrelation of synaptic weights of individual synapses on definite
1 Appendix have been written in cooperation with Dr. I. Smirnitskaja
4.1 Cerebellar Microanatomy The data about the precise cell number and quantitative data on interneuronal connections are not available now for any real cerebellum or cerebellar type. These data probably can be obtained by a complete reconstruction of a macroscopically significant cortex region (e.g. of the submillimeter size) on the basis of the electron microscopy of thin serial slices. The need for such knowledge is crucial regarding the main model parameters.
413
For some types of synaptic weight distributions, the expressions for Pfm and Pmiss are rather simple. In particular, for a, = 0, a,, = 1 we have Pfm =
1 --
(A1)
Fm/:(h ) ,
Pmiss = F,,,/t(h) ,
where h is the Purkinje cell threshold. When the unlearnt synaptic weight is - 1 and the learnt weight is + I / N , only one excited unlearnt synapse blocks the Purkinje cell firing. The Purkinje cell is excited when the number of excitation units arriving to it exceeds h. In this case, Pmi~scoincides with the previous o n e : Pmis~=Fm/t{h/am}. The Pfm can be estimated as follows. Let p be the fraction of the learnt Purkinje cell synapses and L be the number of granule cells excited by an unlearnt input event. Then, pL would be the probability of all the L excited granule cells to possess only learnt synapses. Hence, the probability of the arriving at the Purkinje cell of more than h learnt elementary actions and no unlearnt will be Pfm = E pL. F(L).
(A2)
L >>.h
Adopting an Gaussian approximation, we have
where Sgx
J'x, 0,
when when
x>0, x>Pc" The same approximation yields N(0)=N/" 9(1 - Pfm)" For the residual uncertainty we finally have
nres=,#" H(pe)-n 9((1 --Pmiss)" ld((1 -- Pmiss)/P(1)) + Pmi~ "ld (PmiJP(0))) 9
(A5)
The amount of the information, extractable from the scheme, is given by the expression Iext - -
Ho - H r , = n. ((1 -
Pmiss)
"ld((1-Pmiss)/P(1)) + Pm~s~"ld(Pm~JP(0))),
(A6)
when Pe 4 PYre(A6) gives:
if n is small for the absence of identical subevents of the size Eo in different events of the size L0. In the case described, the probability of any event of the size Eo to (',o) , be learnt is p~ = n. Eo /,4r. This formula is valid if p ) ~ 1. As it was described above, every event of the size Eo, according to the last formula, posseses an uncertainty /-/(p)). Different Eo-sized events are not independently become learnt but are linked in groups consisting of (L~ \ E o ] events9 This fact means that the uncertainty, linked to each subevent, is less than/-/(p)). Suppose it may be expressed as t/./-/(p), where O < t/< 1. The value of t/is determined from a natural condition: W ' . t/. H(p)) = Ho. Now, we can follow the line presented above in No. 2 of the Appendix. We define values ~'(t~/0) and ~'(f/1) for the events of the size Lo, grounding on empirically derived Pmi,s and Pfm" The formulas coincide with the corresponding expressions for full events9 The residue uncertainty is Hres = t/. X ' . (P(1). H(~'(?/1)) + e(0).
I~xt = n. ((1 -Pmi.)" ld(1/pfm) + Pmi~ "ld (1/(1 -- Pfm)) -- H(Pm~.)) 9 Finally for
Pfrn '~ 1 w e
(A7)
have
Iext= n-((1 --Pmi.)" ld(1/pfm)-H(Pmi.)) 9
And the information content is I = Ho - Hr.
e,.)
(A7')
3 Information Contents of the Mart's Memory Unit Tested by Subevents of Learnt Events. Learning is
~ n
events are learnt, ( n" \Eo,]/I ( L ~ subevents bec~
learnt'
9
ld((N~
Lo
n (Eo)))
\\Eo/
performed by the Lo-sized events and the information recall by the Eo-sized events, Eo >P~, then
,,((::) e,n) I=n.
i e/(n
9 Lo
9((1 -p__). ld(1/pem) -4-Pmi~ "ld (1/( 1 - Pfm)) -- H ( P m J ) .
(A 8 3
The main difference between (A8) and (A6) is in the factor, which accounts for the fact that information recall is achieved by the events of a smaller size than in the process of information learning. This factor always exceeds 1. It is, however, evident: one learnt full event produces multiple learnt subevents and because of it, rather higher error probabilities are possible at the same information content as in the case of full events.
415
References Dunin-Barkowski, W.L.: Information processes in neural structures. Moscow: Nauka 1978 (in Russian) Dunin-Barkowski, W.L., Larionova, N.P.: Computer simulation ofa cerebellar cortex compartment. I. General principles and properties of a neural net. Biol. Cybern. 51, 399-406 (1985) Noda, H., Suzuki, D.: The role of the flocculus of the monkey in fixation and smooth pursuit eye movements. J. Physiol. 294, 335-348 (1979) Pellionisz, A., Szentagothai, J.: Dynamic single unit simulation of a realistic cerebellar network model. Brain Res. 49, 83-99 (1973) Smolyaninov, V.V.: Some special veatures of organization of the cerebellar cortex. In: Models of the structural-functional
organization of certain biological systems, pp. 251-325. Cambridge, MA: MIT Press 1971 Torioka, T., Koga, K.: Pattern separability of a three-layered random nerve network with inhibitory connections. Electron. Commun. Jpn. 60A, 34-41 (1977) Received: January 2, 1985 Dr. W. L. Dunin-Barkowski Dr. N. P. Larionova Institute for Problems of Information Transmission USSR Academy of Sciences Moscow USSR
