Psychological Review
Vol. 65, No. 6, 19S8

THE PERCEPTRON: A PROBABILISTIC MODEL FOR
INFORMATION STORAGE AND ORGANIZATION
IN THE BRAIN 1
F. ROSENBLATT
Cornell Aeronautical Laboratory

If we are eventually to understandthe capability of higher organisms forperceptual recognition, generalization,
recall, and thinking, we must firsthave answers to three fundamental
questions:
1. How is information about thephysical world sensed, or detected, by
the biological system?
2. In what form is information
stored, or remembered?
3. How does information containedin storage, or in memory, influence
recognition and behavior?
The first of these questions is in theprovince of sensory physiology, and is
the only one for which appreciable
understanding has been achieved.
This article will be concerned primarily with the second and third
questions, which are still subject to avast amount of speculation, and where
the few relevant facts currently supplied by neurophysiology have not yet
been integrated into an acceptable
theory.
With regard to the second question,
two alternative positions have been
maintained. The first suggests thatstorage of sensory information is in
the form of coded representations orimages, with some sort of one-to-one
mapping between the sensory stimulus
1
The development of this theory has been
carried out at the Cornell Aeronautical Laboratory, Inc., under the sponsorship of the
Office of Naval Research, Contract Nonr2381(00). This article is primarily'an adaptation of material reported in Ref. IS, whichconstitutes the first full report on the program.

386

and the stored pattern. According tothis hypothesis, if one understood thecode or "wiring diagram" of the nervous system, one should, in principle,
be able to discover exactly what anorganism remembers by reconstructing the original sensory patterns fromthe "memory traces" which they have
left, much as we might develop a
photographic negative, or translate
the pattern of electrical charges in the"memory" of a digital computer.
This hypothesis is appealing in its
simplicity and ready intelligibility,
and a large family of theoretical brain
models has been developed around theidea of a coded, representational memory (2, 3, 9, 14). The alternative approach, which stems from the tradition of British empiricism, hazards theguess that the images of stimuli may
never really be recorded at all, andthat the central nervous system
simply acts as an intricate switching
network, where retention takes theform of new connections, or pathways,
between centers of activity. In manyof the more recent developments ofthis position (Hebb's "cell assembly,"and Hull's "cortical anticipatory goal
response," for example) the "responses" which are associated tostimuli may be entirely contained
within the CNS itself. In this casethe response represents an "idea"
rather than an action. The important feature of this approach is that
there is never any simple mapping of
the stimulus into memory, according
to some code which would permit its
later reconstruction. Whatever in-THE PERCEPTRON
formation is retained must somehow
be stored as a preference for a particular response; i.e., the informationis contained in connections or associations rather than topographic representations. (The term response, forthe remainder of this presentation,
should be understood to mean any
distinguishable state of the organism,
which may or may not involve externally detectable muscular activity.
The activation of some nucleus of cells
in the central nervous system, forexample, can constitute a response,
according to this definition.)
Corresponding to these two positions on the method of information
retention, there exist two hypotheseswith regard to the third question, themanner in which stored information
exerts its influence on current activity.
The "coded memory theorists" areforced to conclude that recognition of
any stimulus involves the matchingor systematic comparison of the contents of storage with incoming sensory patterns, in order to determine
whether the current stimulus has been
seen before, and to determine the appropriate response from the organism.
The theorists in the empiricist tradition, on the other hand, have essentially combined the answer to thethird question with their answer to thesecond: since the stored information
takes the form of new connections, ortransmission channels in the nervous
system (or the creation of conditions
which are functionally equivalent to
new connections), it follows that the
new stimuli will make use of these newpathways which have been created,
automatically activating the appropriate response without requiring any
separate process for their recognition
or identification.
The theory to be presented here
takes the empiricist, or "connectionist"'
position with regard to these ques-387

tions. The theory has been developed
for a hypothetical nervous system, or
machine, called a perceptron. The
perceptron is designed to illustratesome of the fundamental properties ofintelligent systems in general, without
becoming too deeply enmeshed in thespecial, and frequently unknown, conditions which hold for particular biological organisms. The analogy between the perceptron and biologicalsystems should be readily apparent to
the reader.
During the last few decades, thedevelopment of symbolic logic, digitalcomputers, and switching theory has
impressed many theorists with thefunctional similarity between a neuron
and the simple on-off units of which
computers are constructed, and has
provided the analytical methods necessary for representing highly complex
logical functions in terms of suchelements. The result has been aprofusion of brain models whichamount simply to logical contrivances
for performing particular algorithms(representing "recall," stimulus comparison, transformation, and various
kinds of analysis) in response to
sequences of stimuli—e.g., Rashevsky(14), McCulloch (10), McCulloch &Pitts (11), Culbertson (2), Kleene(8), and Minsky (13). A relatively
small number of theorists, like Ashby(1) and von Neumann (17, 18), havebeen concerned with the problems of
how an imperfect neural network,
containing many random connections,can be made to perform reliably thosefunctions which might be represented
by idealized wiring diagrams. Unfortunately, the language of symbolic
logic and Boolean algebra is less well
suited for such investigations. The
need for a suitable language for themathematical analysis of events insystems where only the gross organization can be characterized, and the388

F. ROSENBLATT

precise structure is unknown, has led ogous machines, have generally been
the author to formulate the current less exact in their formulations and far
model in terms of probability theory from rigorous in their analysis, so that
it is frequently hard to assess whether
rather than symbolic logic.
The theorists referred to above were or not the systems that they describe
chiefly concerned with the question of could actually work in a realistic nervhow such functions as perception and ous system, and what the necessary
recall might be achieved by a deter- and sufficient conditions might be.
ministic physical system of any sort, Here again, the lack of an analytic
rather than how this is actually done language comparable in proficiency toby the brain. The models which have the Boolean algebra of the network
been produced all fail in some im- analysts has been one of the main
portant respects (absence of equi- obstacles. The contributions of thispotentiality, lack of neuroeconomy, group should perhaps be considered as
excessive specificity of connections suggestions of what to look for andand synchronization requirements, investigate, rather than as finishedunrealistic specificity of stimuli suffi- theoretical systems in their own right.
cient for cell firing;, postulation of Seen from this viewpoint, the mostvariables or functional features with suggestive work, from the standpoint
no known neurological correlates, etc.) of the following theory, is that ofto correspond to a biological system. Hebb and Hayek.
The position, elaborated by Hebb
The proponents of this line of approach have maintained that, once it (7), Hayek (6), Uttley (16), andhas been shown how a physical Ashby (1), in particular, upon which
system of any variety might be made the theory of the perceptron is based,
to perceive and recognize stimuli, or can be summarized by the following
perform other brainlike functions, it assumptions:would require only a refinement or
1. The physical connections of themodification of existing principles to
nervous system which are involved inunderstand the working of a more learning and recognition are not idenrealistic nervous system, and to elimtical from one organism to another.
inate the shortcomings mentioned
At birth, the construction of the mostabove. The writer takes the position,
important networks is largely random,on the other hand, that these short- subject to a minimum number ofcomings are such that a mere refinegenetic constraints.
ment or improvement of the principles
2. The original system of connectedalready suggested can never account cells is capable of a certain amount offor biological intelligence; a difference
plasticity; after a period of neuralin principle is clearly indicated. The activity, the probability that a stimtheory of statistical separability (Cf. ulus applied to one set of cells will15), which is to be summarized here, cause a response in some other set isappears to offer a solution in principle
likely to change, due to some relato all of these difficulties.
tively long-lasting changes in theThose theorists—Hebb (7), Milner neurons themselves.
(12), Eccles (4), Hayek (6)—who
3. Through exposure to a large
have been more directly concerned sample of stimuli, those which arewith the biological nervous system most "similar" (in some sense which
and its activity in a natural environ- must be defined in terms of thement, rather than with formally'anal- particular physical system) will tend

THE PEECEPTRON
to form pathways to the same sets of
responding cells. Those which aremarkedly "dissimilar" will tend todevelop connections to different sets of
responding cells.
4. The application of positive and/or negative reinforcement (or stimuli
which serve this function) may facilitate or hinder whatever formation ofconnections is currently in progress.
5. Similarity, in such a system, isrepresented at some level of the nervous system by a tendency of similar
stimuli to activate the same sets ofcells. Similarity is not a necessary
attribute of particular formal or geometrical classes of stimuli, but depends on the physical organization of
the perceiving system, an organization which evolves through interaction
with a given environment. The
structure of the system, as well as theecology of the stimulus-environment,
will affect, and will largely determine,the classes of "things" into which theperceptual world is divided.
THE ORGANIZATION OF A PERCEPTRON
The organization of a typical photoperceptron (a perceptron responding
to optical patterns as stimuli) is shownin Fig. 1. The rules of its organization are as follows:
1. Stimuli impinge on a retina of
sensory units (S-points), which areassumed to respond on an all-ornothing basis, in some models, or witha pulse amplitude or frequency proportional to the stimulus intensity, inother models. In the models considered here, an all-or-nothing response will be assumed.
2. Impulses are transmitted to a setof association cells (A-units) in a"projection area" (Ai). This projection area may be omitted in somemodels, where the retina is connected
directly to the association area (An).

389

FIG. 1. Organization of a perceptron.

The cells in the projection area each
receive a number of connections fromthe sensory points. The set of Spoints transmitting impulses to a particular A-unit will be called the origin
points of that A-unit. These origin
points may be either excitatory or inhibitory in their effect on the A-unit.
If the algebraic sum of excitatory andinhibitory impulse intensities is equal
to or greater than the threshold (6) of
the A-unit, then the A-unit fires, again
on an all-or-nothing basis (or, in somemodels, which will not be considered
here, with a frequency which depends
on the net value of the impulses
received). The origin points of theA-units in the projection area tend to
be clustered or focalized, about some
central point, corresponding to each
A-unit. The number of origin points
falls off exponentially as the retinal
distance from the central point for
the A-unit in question increases.
(Such a distribution seems to be supported by physiological evidence, andserves an important functional purpose in contour detection.)
3. Between the projection area andthe association area (An), connections
are assumed to be random. That is,each A-unit in the An set receives
some number of fibers from origin
points in the AI set, but these origin
points are scattered at random
throughout the projection area.
Apart from their connection distribution, the An units are identicalwith the AI units, and respond undersimilar conditions.
4. The "responses," Ri, R^, . . . ,R n are cells (or sets of cells) which

390

F. ROSENBLATT

respond in much the same fashion as
the A-units. Each response has atypically large number of origin points
located at random in the An set. The
set of A-units transmitting impulses
to a particular response will be called
the source-set for that response.
(The source-set of a response is identical to its set of origin points in the
A-system.) The arrows in Fig. 1
indicate the direction of transmission
through the network. Note that up
to An all connections are forward, andthere is no feedback. When we cometo the last set of connections, between
An and the R-units, connections are
established in both directions. The
rule governing feedback connections,in most models of the perceptron, canbe either of the following alternatives:(a) Each response has excitatory
feedback connections to the cells in its
own source-set, or(&) Each response has inhibitory
feedback connections to the complement of its own source-set (i.e., it tends
to prohibit activity in any association
cells which do not transmit to it).
The first of these rules seems moreplausible anatomically, since the Runits might be located in the same
cortical area as their respective source-

flHOKfflt LIHCt >W
IHHtllTOMV
COHKECTIOHI

FIG. 2A. Schematic representation ofconnections in a simple perceptron.

OWHWTMV OMNICTIOH• EMITITMV CONNtcriON

FIG. 2B. Venn diagram of the same perceptronf (shading shows active sets for RI
response).

sets, making mutual excitation between the R-units and the A-units of
the appropriate source-set highly
probable. The alternative rule (6)leads to a more readily analyzed system, however, and will therefore be
assumed for most of the systems to beevaluated here.
Figure 2 shows the organization ofa simplified perceptron, which affords
a convenient entry into the theory ofstatistical separability. After thetheory has been developed for thissimplified model, we will be in a better
position to discuss the advantages ofthe system in Fig. 1. The feedback
connections shown in Fig. 2 are inhibitory, and go to the complementof the source-set for the response fromwhich they originate; consequently,
this system is organized according toRule b, above. The system shown
here has only three stages, the first
association stage having been eliminated. Each A-unit has a set of
randomly located origin points in theretina. Such a system will form similarity concepts on the basis of coincident areas of stimuli, rather than by
the similarity of contours or outlines.
While such a system is at a disadvantage in many discrimination experiments, its capability is still quite
impressive, as will be demonstrated
presently. The system shown in Fig.
2 has only two responses, but there isclearly no limit on the number thatmight be included.
The responses in a system organizedin this fashion are mutually exclusive.
If RI occurs, it will tend to inhibit Rs,
and will also inhibit the source-set forR2. Likewise, if Ra should occur, it
will tend to inhibit RI. If the total
impulse received from all the A-units
in one source-set is stronger or more
frequent than the impulse received
by the alternative (antagonistic) response, then the first response will

391

THE PERCEPTRON
tend to gain an advantage over theother, and will be the one which
occurs. If such a system is to becapable of learning, then it must be
possible to modify the A-units or theirconnections in such a way that stimuli
of one class will tend to evoke astronger impulse in the Ri source-set
than in the Ra source-set, while
stimuli of another (dissimilar) classwill tend to evoke a^stronger impulsein the Ra source-set than in the Ri
source-set.
It will be assumed that the impulses
delivered by each A-unit can becharacterized by a value, V, which
may be an amplitude, frequency,
latency, or probability of completingtransmission. If an A-unit has a highvalue, then all of its output impulses
are considered to be more effective,

more potent, or more likely to arriveat their endbulbs than impulses froman A-unit with a lower value. The
value of an A-unit is considered to bea fairly stable characteristic, probably
depending on the metabolic conditionof the cell and the cell membrane, but
it is not absolutely constant. It is
assumed that, in general, periods ofactivity tend to increase a cell's value,
while the value may decay (in somemodels) with inactivity. The mostinteresting models are those in whichcells are assumed to compete for metabolic materials, the more active cells
gaining at the expense of the lessactive cells. In such a system, if
there is no activity, all cells will tend
to remain in a relatively constantcondition, and (regardless of activity)
the net value of the system, taken in

TABLE 1
COMPARISON or LOGICAL CHARACTERISTICS OF a, /3, AND 7 SYSTEMS
a-System(UncompensatedGain System)/3-System(Constant Feed
System)7-System(Parasitic Gain
System)Total value-gain of source set per reinforcement

Nar

K

0

AV for A-units active for 1 unit of time

+1

K/Nar

+1

AV for inactive A-units outside of dominant set

0

K/NA,

0

A V for inactive A-units of dominant set

0

0

-N*r
NAr-NarMean value of A-system

Increases with number Increases withtimeof reinforcements

Difference between mean values of Proportional to differsource-sets
ences of reinforcement frequency(»,„—»,„)

0Constant0Note: In the /3 and 7 systems, the total value-change for any A-unit will be the sum of the AV'sfor all source-sets of which it is a member.
N»r = Number of active units in source-set
NAT — Total number of units in source-set«,„ = Number of stimuli associated to response TJ
K = Arbitrary constant

392

F. ROSENBLATT

its entirety, will remain constant at
all times. Three types of systems,
which differ in their value dynamics,
have been investigated quantitatively.
Their principal logical features arecompared in Table 1. In the alphasystem, an active cell simply gains anincrement of value for every impulse,and holds this gain indefinitely. In
the beta system, each source-set isallowed a certain constant rate of gain,
the increments being apportioned
among the cells of the source-set inproportion to their activity. In the
gamma system, active cells gain invalue at the expense of the inactive
cells of their source-set, so that the
total value of a source-set is always
constant.
For purposes of analysis, it is convenient to distinguish two phases in
the response of the system to a stimulus (Fig. 3). In the predominant
phase, some proportion of A-units(represented by solid dots in the
figure) responds to the stimulus, but
the R-units are still inactive. This
phase is transient, and quickly gives
way to the postdominant phase, inwhich one of the responses becomes
active, inhibiting activity in the com-

FIG. 3A. Predominant phase. Inhibitory connections
are not shown. Solid black units are active.

FIG. 3B. Postdominant phase. Dominant subset
suppresses rival' sets. Inhibitory connections shown
only for Ri.

FIG. 3. Phases of response to a stimulus,

plement of its own source-set, andthus preventing the occurrence of any
alternative response. The response
which happens to become dominant isinitially random, but if the A-units are
reinforced (i.e., if the active units areallowed to gain in value), then when
the same stimulus is presented again
at a later time, the same response will
have a stronger tendency to recur, andlearning can be said to have takenplace.
ANALYSIS OF THE PREDOMINANT
PHASE
The perceptrons considered here
will always assume a fixed threshold,6, for the activation of the A-units.
Such a system will be called a fixedthreshold model, in contrast to a continuous transducer model, where theresponse of the A-unit is some continuous function of the impinging
stimulus energy.
In order to predict the learning
curves of a fixed-threshold perceptron,
two variables have been found to beof primary importance. They aredefined as follows :Pa = the expected proportion of Aunits activated by a stimulus of agiven size,
PC = the conditional probability
that an A-unit which responds to agiven stimulus, Si, will also respondto another given stimulus, 82It can be shown (Rosenblatt, IS) thatas the size of the retina is increased,
the number of S-points (Na} quickly
ceases to be a significant parameter,
and the values of Pa and Pc approachthe value that they would have for aretina with infinitely many points.
For a large retina, therefore, theequations are as follows :(1)

THE PERCEPTRON

393

where

and

P(e,i) =L — proportion of the S-points illuminated by the first stimulus, Si,
which are not illuminated by
S2
G — proportion of the residual S-set(left over from the first stimulus) which is included in thesecond stimulus (82).

X

andR = proportion of S-points activatedby the stimulus
x — number of excitatory connections to each A-unit
y = number of inhibitory connections to each A-unit
0 = threshold of A-units.
(The quantities e and i are the excitatory and inhibitory components ofthe excitation received by the A-unit
from the stimulus. If the algebraic
sum a = e + i is equal to or greaterthan 6, the A-unit is assumed to respond.)xjy•* e = " ~ 2~i 2-i
x—0

e

i

2—i 2-

y~~^£P(«,t,J.,/<,«.,«<)(e -- J, + /,- + ge - «,- > 0)

where

X
X

(2)

>

x (/\x -s^ e\j/ G"«(I - G) *-«-<'•

x (';')G.,(1-G).-.,The quantities R, L, and G specify thetwo stimuli and their retinal overlap.
le and /,• are, respectively, the numbersof excitatory and inhibitory origin
points "lost" by the A-unit whenstimulus Si is replaced by 82; ge andgi are the numbers of excitatory andinhibitory origin points "gained"
when stimulus Si is replaced by 82.
The summations in Equation 2 arebetween the limits indicated, subject
to the side condition e — i — I, + l{+g, - gi > 0.
Some of the most important characteristics of Pa are illustrated in Fig.
4, which shows Pa as a function of theretinal area illuminated (R). Notethat Pa can be reduced in magnitude
by either increasing the threshold, 6,
or by increasing the proportion of inhibitory connections (y). A comparison of Fig. 4b and 4c shows that if theexcitation is about equal to the inhibition, the curves for Pa as a functionof R are flattened out, so that there islittle variation in Pa for stimuli ofdifferent sizes. This fact is of great
importance for systems which require
Pa to be close to an optimum value inorder to perform properly.
The behavior of Pc is illustrated inFig. 5 and 6. The curves in Fig. 5 canbe compared with those for Pa in Fig.
4. Note that as the threshold is increased, there is an even sharper reduction in the value of Pc than was thecase with Pa. Pc also decreases as the
proportion of inhibitory connections
increases, as does Pa. Fig. 5, which isF. ROSENBLATT

394(•] EFFECT OF INHIBITORYEXCITATORY MIXTURE. 6 " I

0
.1
.2
.3
.1
.5
PROPORTION OF S-POIHTS ILLUMINATED(K)(b) VARIATION WITH 6
FOR X • 10, V * 0

.1

.2

.3

.it

(c) VARIATION WITH M AHD 0
FOR MIXTURES ABOUT.60J
INHIBITORY. SOLID LINESARE FOR X • 5, Y » 5.

.B

FIG. 4. P0 as function of retinal area illuminated.

calculated for nonoverlapping stimuli,
illustrates the fact that Pc remains
greater than zero even when the stimuli are completely disjunct, and illuminate no retinal points in common. In
Fig. 6, the effect of varying amountsof overlap between the stimuli isshown. In all cases, the value of Pc
goes to unity as the stimuli approach
perfect identity. For smaller stimuli(broken line curves), the value of Pc
is lower than for large stimuli. Similarly, the value is less for high thresholds than for low thresholds. The
minimum value of Pc will be equal to
Pcmin = (1 - L)'(l - G)». (3)In Fig. 6, Pemin corresponds to thecurve for 6 = 10. Note that underthese conditions the probability that
the A-unit responds to both stimuli(Pc) is practically zero, except for
stimuli which are quite close to
identity. This condition can be of
considerable help in discrimination
learning.

MATHEMATICAL ANALYSIS
OF LEARNING IN THE
PERCEPTRON
The response of the perceptron in
the predominant phase, where somefraction of the A-units (scatteredthroughout the system) responds tothe stimulus, quickly gives way to thepostdominant response, in which activity is limited to a single source-set,
the other sets being suppressed. Two
possible systems have been studied forthe determination of the "dominant"response, in the postdominant phase.
In one (the mean-discriminating system, or ju-system), the response whose
inputs have the greatest mean value
responds first, gaining a slight advantage over the others, so that it quickly
becomes dominant. In the secondcase (the sum-discriminating system,or S-system), the response whose inputs have the greatest net value gains
an advantage. In most cases, systems which respond to mean values
have an advantage over systems which
respond to sums, since the means are395

THE PERCEPTEON

.1

.2
.3
.4
.5
R (RETINAL AREA ILLUMINATED)

FIG. 5. Pc as a function of R,
for nonoverlapping stimuli.

less influenced by random variationsin Pa from one source-set to another.
In the case of the -/-system (see Table1), however, the performance ofthe ^-system and S-system become
identical.
We have indicated that the perceptron is expected to learn, or to form
associations, as a result of the changes
in value that occur as a result of theactivity of the association cells. In
evaluating this learning, one of two
types of hypothetical experiments canbe considered. In the first case, theperceptron is exposed to some seriesof stimulus patterns (which might bepresented in random positions on theretina) and is "forced" to give thedesired response in each case. (This
forcing of responses is assumed to bea prerogative of the experimenter. In
experiments intended to evaluate
trial-and-error learning, with moresophisticated perceptrons, the experimenter does not force the system torespond in the desired fashion, but
merely applies positive reinforcement
when the response happens to be correct, and negative reinforcement when
the response is wrong.) In evaluating
the learning which has taken place
during this "learning series," theperceptron is assumed to be "frozen"
in its current condition, no further
value changes being allowed, and thesame series of stimuli is presented
again in precisely the same fashion, sothat the stimuli fall on identical positions on the retina. The probability
that the perceptron will show a bias
towards the "correct" response (theone which has been previously reinforced during the learning series) inpreference to any given alternative
response is called Pr, the probabilityof correct choice of response between
two alternatives.
In the second type of experiment, alearning series is presented exactly as
before, but instead of evaluating theperceptron's performance using thesame series of stimuli which wereshown before, a new series is presented, in which stimuli may be drawn
from the same classes that were previ-(PROPORTION OF 0¥E«L«P BETMEEK STIMULI)

FIG. 6. Pc as a function of C. X - 10,
Y = 0. Solid lines: R = .5; broken lines:
R = .2.

396

F. ROSENBLATT

ously experienced, but are not necessarily identical. This new test series
is assumed to be composed of stimuli
projected onto random retinal positions, which are chosen independentlyof the positions selected for the learning series. The stimuli of the test
series may also differ in size or rotational position from the stimuli whichwere previously experienced. In thiscase, we are interested in the probability that the perceptron will givethe correct response for the class ofstimuli which is represented, regardless of whether the particular stimulus
has been seen before or not. This
probability is called Pt, the probability of correct generalization. As
with Pr, Pg is actually the probability
that a bias will be found in favor of theproper response rather than any one
alternative ; only one pair of responses
at a time is considered, and the factthat the response bias is correct in one
pair does not mean that there may
not be other pairs in which the bias
favors the wrong response. The probability that the correct response will
be preferred over all alternatives isdesignated PR or Pa.
In all cases investigated, a single
general equation gives a close approximation to Pr and Pg, if the appropriate constants are substituted.
This equation is of the form :Ra after n,r stimuli have been shownfor each of the two responses, during
the learning period. N, is the numberof "effective" A-units in each sourceset ; that is, the number of A-units ineither source-set which are not connected in common to both responses.
Those units which are connected incommon contribute equally to bothsides of the value balance, and consequently do not affect the net bias
towards one response or the other.
Nar is the number of active units in asource-set, which respond to the test
stimulus, St-P(Nar > 0) is the probability that at least one of the Ne
effective units in the source-set of thecorrect response (designated, by convention, as the Ri response) will be
activated by the test stimulus, St.
In the case of Pg, the constant c2 isalways equal to zero, the other threeconstants being the same as for Pr.
The values of the four constants
depend on the parameters of thephysical nerve net (the perceptron)
and also on the organization of thestimulus environment.
The simplest cases to analyze arethose in which the perceptron is shown
stimuli drawn from an "ideal environment," consisting of randomly placed
points of illumination, where there is
no attempt to classify stimuli according to intrinsic similarity. Thus, in a
typical learning experiment, we might
P = P(^ a r >0).<£(Z)(4) show the perceptron 1,000 stimuli
made up of random collections of
where
illuminated retinal points, and weP(Nar > 0) = 1 - (1 - P.)*.
might arbitrarily reinforce Ri as the"correct" response for the first 500<t>(Z) = normal curve integralofthese, and R.2 for the remaining 500.
from — oo to Z
This environment is "ideal" only inandthe sense that we speak of an ideal gas
in physics; it is a convenient artifact
c4n,
for purposes of analysis, and does not
If Ri is the "correct" response, and R 2 lead to the best performance from theis the alternative response under con- perceptron. In the ideal environsideration, Equation 4 is the prob- ment situation, the constant c\ isability that Rj, will be preferred over always equal to zero, so that, in the397

THE PERCEPTRON
case of Pg (where c2 is also zero), thevalue of Z will be zero, and Pg cannever be any better than the random
expectation of 0.5. The evaluationof Pr for these conditions, however,throws some interesting light on the
differences between the alpha, beta,
and gamma systems (Table 1).
First consider the alpha system,
which has the simplest dynamics ofthe three. In this system, wheneveran A-unit is active for one unit oftime, it gains one unit of value. We
will assume an experiment, initially,in which N,r (the number of stimuli
associated to each response) is constant for all responses. In this case,for the sum system,

(5)

100

where u> = the fraction of responses
connected to each A-unit. If thesource-sets are disjunct, w = I/NR,
where NR is the number of responses
in the system. For the ^-system,(6)

The reduction of c3 to zero gives the^-system a definite advantage over the
S-system. Typical learning curves
for these systems are compared inFig. 7 and 8. Figure 9 shows theeffect of variations in Pa upon theperformance of the system.
If n,r, instead of being fixed, istreated as a random variable, so thatthe number of stimuli associated toeach response is drawn separatelyfrom some distribution, then the per-1000

10,0000,n (NUMBER OF STIMULI ASSOCIATED TO EACH RESPONSE)

FIG. 7. P,( 2) as function of an,, for discrete subsets.
(a>« = 0, Pa = .005. Ideal environment assumed.)

100,000

F. ROSENBLATT

398

10,000100,000

FIG. 8. Prdit as function of an,. (For Pa — .07, wc = 0. Ideal environment assumed.)

formance of the a-system is considerably poorer than the above equations
indicate. Under these conditions, theconstants for the //-system are
Ci — 0C2 = 1 - Par MT B -I)«

LC^ —7Vfl-2

]

^XJ

(7)

2(1 - Pa)where
q = ratio of 0*,, to n,,
NR = number of responses in the system
NA = number of A-units in the systemco,. = proportion of A-units common
to RX and R2.
For this equation (and any others inwhich n,r is treated as a randomvariable), it is necessary to define n,rin'. Equation 4 as the expected value
off this variable, over the set of all
responses.

For the /3-system, there is an evengreater deficit in performance, due tothe fact that the net value continues
to grow regardless of what happens
to the system. The large net valuesof the subsets activated by a stimulus
tend to amplify small statistical differences, causing an unreliable performance. The constants in this case(again for the /u-system) are
ci = 0c, = (1 - Pa)Ne
c, = 2(PaNequNB*)*c« = 2(1 -(8)In both the alpha and beta systems,
performance will be poorer for thesum-discriminating model than for themean-discriminating case. In the
gamma-system, however, it can beshown thatP,(s) = PK/O; i.e., it makes
no difference in performance whether
the S-system or ^-system is used.
Moreover, the constants for the ysystem, with variable nsr, are identicalto the constants for the alpha jt-sys-

399

THE PERCEPTRON

FIG. 9. P,tf) as function of P*.

(For n,T — 1,000, u, = 0.

tern, with nsr fixed (Equation 6).
The performance of the three systems
is compared in Fig. 10, which clearly

Ideal environment assumed.)demonstrates the advantage of the
7-system.
Let us now replace the "ideal en-

10,000

FIG. 10. Comparison of a, /3, and 7 systems, for variable «»,(NR = 100, <mr, = .5*,,, NA = 10,000, Pa = ,07, w = ,2).

400

F. ROSENBLATT

vironment" assumptions with a model stimuli drawn at random from all
for a "differentiated environment," in other classes in the environment.
which several distinguishable classes
If Pcll > Pa > PC12, the limitingof stimuli are present (such as squares, performance of the perceptron (PBoo)
circles, and triangles, or the letters of will be better than chance, and learnthe alphabet). If we then design an ing of some response, RI, as the proper
experiment in which the stimuli asso- "generalization response" for memciated to each response are drawn from bers of Class 1 should eventuallya different class, then the learning occur. If the above inequality is not
curves of the perceptron are drasti- met, then improvement over chancecally altered. The most important performance may not occur, and thedifference is that the constant c\ (the Class 2 response is likely to occur
coefficient of nsr in the numerator of Z) instead. It can be shown (IS) thatis no longer equal to zero, so that for most simple geometrical forms,
Equation 4 now has a nonrandom which we ordinarily regard as "simiasymptote. Moreover, in the form lar," the required inequality can befor P, (the probability of correct met, if the parameters of the systemgeneralization), where c% = 0, the are properly chosen.
quantity Z remains greater than zero,The equation for Pr, for the sumand Pa actually approaches the same discriminating version of an a-percepasymptote as Pr. Thus the equation tron, in a differentiated environment
for the perceptron's performance after where n, is fixed for all responses, will
infinite experience with each class of have ther following expressions for the
stimuli is identical for PT and Pg :four coefficients:

C 2 =PJV,(1-P 0 1 1 )This means that in the limit it makes
no difference whether the perceptron has
seen a particular test stimulus before ornot; if the stimuli are drawn from adifferentiated environment, the performance will be equally good in either case.
In order to evaluate the performance of the system in a differentiatedenvironment, it is necessary to define
the quantity PCap. This quantity isinterpreted as the expected value of
PC between pairs of stimuli drawn at
random from classes a and j3. In
particular, PcU is the expected valueof Pc between members of the sameclass, and Peis is the expected value of
PC between an Si stimulus drawn fromClass 1 and an S2 stimulus drawn fromClass 2. Pclx is the expected value of
Pc between members of Class 1 and

r=l,2

(10)

r=»l,2

whereir) and a?(Pc\x) represent thevariance of Pcir and Pc\x measured over the set of possible
test stimuli, St, and401

THE PERCEPTSON\ — \Pcn~ PM)iJ and crf(Pcix) represent thevariance of Pcir and P^ measured over the set of all A-units,e = covariance of PclrPclx, which isassumed to be negligible.
The variances which appear in these
expressions have not yielded, thus far,to a precise analysis, and can betreated as empirical variables to be
determined for the classes of stimuli
in question. If the sigma is set equal
to half the expected value of the variable, in each case, a conservative
estimate can be obtained. When thestimuli of a given class are all of thesame shape, and uniformly distributed
over the retina, the subscript s variances are equal to zero. Paw will be
represented by the same set of coefficients, except for c2, which is equal tozero, as usual.
For the mean-discriminating system, the coefficients are:X[cr/(Prtr
(ID
C

4=

-i p XT [_Pelrr— 1,2 -* a-t'e

Pclr

Some covariance terms, which areconsidered negligible, have been omitted here.
A set of typical learning curves forthe differentiated environment modelis shown in Fig. 11, for the meandiscriminating system. The parameters are based on measurements for a

10,000

FIG. 11. P, and Pg as function of «,r. Parameters based on square-circle discrimination.

402

F. ROSENBLATT

square-circle discrimination problem.
Note that the curves for Pr and Pg
both approach the same asymptotes,
as predicted. The values of these
asymptotes can be obtained by substituting the proper coefficients inEquation 9. As the number of association cells in the system increases,
the asymptotic learning limit rapidly
approaches unity, so that for a systemof several thousand cells, the errors in
performance should be negligible on a
problem as simple as the one illustrated here.
As the number of responses in thesystem increases, the performance becomes progressively poorer, if every
response is made mutually exclusiveof all alternatives. One method of
avoiding this deterioration (describedin detail in Rosenblatt, 15) is throughthe binary coding of responses. In
this case, instead of representing 100
different stimulus patterns by 100
distinct, mutually exclusive responses,
a limited number of discriminating
features is found, each of which can be
independently recognized as being
present or absent, and consequently
can be represented by a single pair ofmutually exclusive responses. Givenan ideal set of binary characteristics(such as dark, light; tall, short;straight, curved; etc.), 100 stimulus
classes could be distinguished by theproper configuration of only sevenresponse pairs. In a further modification of the system, a single response iscapable of denoting by its activity or
inactivity the presence or absence ofeach binary characteristic. The efficiency of such coding depends on thenumber of independently recognizable"earmarks" that can be found todifferentiate stimuli. If the stimulus
can be identified only in its entirety
and is not amenable to such analysis,
then ultimately a separate binary
response pair, or bit, is required todenote the presence or absence of eachstimulus class (e.g., "dog" or "notdog"), and nothing has been gained
over a system where all responses aremutually exclusive.

BIVALENT SYSTEMSIn all of the systems analyzed up to
this point, the increments of value
gained by an active A-unit, as a resultof reinforcement or experience, havealways been positive, in the sense thatan active unit has always gained inits power to activate the responses
to which it is connected. In the
gamma-system, it is true that someunits lose value, but these are always
the inactive units, the active ones
gaining in proportion to their rate ofactivity. In a bivalent system, twotypes of reinforcement are possible(positive and negative), and an active
unit may either gain or lose in value,depending on the momentary state ofaffairs in the system. If the positive
and negative reinforcement can becontrolled by the application of external stimuli, they become essentiallyequivalent to "reward" and "punishment," and can be used in this senseby the experimenter. Under these
conditions, a perceptron appears to becapable of trial-and-error learning. A
bivalent system need not necessarily
involve the application of reward andpunishment, however. If a binarycoded response system is so organizedthat there is a single response or
response-pair to represent each "bit,"or stimulus characteristic that islearned, with positive feedback to its
own source-set if the response is "on,"and negative feedback (in the sensethat active A-units will lose rather
than gain in value) if the response is"off," then the system is still bivalent
in its characteristics. Such a bivalent

403

THE PERCEPTRON
system is particularly efficient in reducing some of the bias effects (preference for the wrong response due togreater size or frequency of its associated stimuli) which plague the alternative systems.
Several forms of bivalent systems
have been considered (15, Chap. VII).
The most efficient of these has thefollowing logical characteristics.
If the system is under a state of
positive reinforcement, then a positive
AV is added to the values of all active
A-units in the source-sets of "on"responses, while a negative AV isadded to the active units in the sourcesets of "off" responses. If the system
is currently under negative reinforcement, then a negative AV is added to
all active units in the source-set of an"on" response, and a positive AV isadded to active units in an "off"source-set. If the source-sets are
disjunct (which is essential for thissystem to work properly), the equation for a bivalent -y-system has thesame coefficients as the monovalenta-system, for the /j-case (Equation 11).
The performance curves for thissystem are shown in Fig. 12, where theasymptotic generalization probability
attainable by the system is plotted forthe same stimulus parameters thatwere used in Fig. 11. This is theprobability that all bits in an «-bitresponse pattern will be correct.
Clearly, if a majority of correct responses is sufficient to identify a stimulus correctly, the performance will bebetter than these curves indicate.
In a form of bivalent system which
utilizes more plausible biological assumptions, A-units may be either
excitatory or inhibitory in their effecton connected responses. A positive
AV in this system corresponds to theincrementing of an excitatory unit,
while a negative AV corresponds tothe incrementing of an inhibitory unit.

Such a system performs similarly tothe one considered above, but can beshown to be less efficient.
Bivalent systems similar to those
illustrated in Fig. 12 have been
simulated in detail in a series of experiments with the IBM 704 computerat the Cornell Aeronautical Laboratory. The results have borne out
the theory in all of its main predictions, and will be reported separately
at a later time.
IMPROVED PERCEPTRONS ANDSPONTANEOUS ORGANIZATIONThe quantitative analysis of perceptron performance in the preceding
sections has omitted any considerationof time as a stimulus dimension. A
perceptron which has no capability
for temporal pattern recognition isreferred to as a "momentary stimulus
perceptron." It can be shown (15)that the same principles of statisticalseparability will permit the perceptron
to distinguish velocities, sound sequences, etc., provided the stimuli
leave some temporarily persistent
trace, such as an altered threshold,

2000

woo

6000

FIG. 12. Pga for a bivalent binary system(same parameters as Fig. 11).

404

F. ROSENBLATT

which causes the activity in the Asystem at time t to depend to somedegree on the activity at time t — 1.
It has also been assumed that theorigin points of A-units are completely
random. It can be shown that by asuitable organization of origin points,
in which the spatial distribution isconstrained (as in the projection area
origins shown in Fig. 1), the A-units
will become particularly sensitive tothe location of contours, and performance will be improved.
In a recent development, which wehope to report in detail in the near
future, it has been proven that if thevalues of the A-units are allowed todecay at a rate proportional to theirmagnitude, a striking new property
emerges: the perceptron becomes capable of "spontaneous" concept formation. That is to say, if the system is
exposed to a random series of stimuli
from two "dissimilar" classes, and allof its responses are automatically reinforced without any regard to whether
they are "right" or "wrong," thesystem will tend towards a stableterminal condition in which (for eachbinary response) the response will be"1" for members of one stimulus class,and "0" for members of the otherclass; i.e., the perceptron will spontaneously recognize the difference
between the two classes. This phenomenon has been successfully demonstrated in simulation experiments,
with the 704 computer.
A perceptron, even with a single
logical level of A-units and responseunits, can be shown to have a numberof interesting properties in the field of
selective recall and selective attention.
These properties generally depend on
the intersection of the source sets fordifferent responses, and are elsewhere
discussed in detail (IS). By combining audio and photo inputs, it ispossible to associate sounds, or audi-

tory "names" to visual objects, and to
get the perceptron to perform suchselective responses as are designatedby the command "Name the object
on the left," or "Name the color ofthis stimulus."
The question may well be raised at
this point of where the perceptron'scapabilities actually stop. We haveseen that the system described is sufficient for pattern recognition, associative learning, and such cognitive sets
as are necessary for selective attention
and selective recall. The system appears to be potentially capable oftemporal pattern recognition, as well
as spatial recognition, involving any
sensory modality or combination ofmodalities. It can be shown thatwith proper reinforcement it will becapable of trial-and-error learning,and can learn to emit ordered sequences of responses, provided its own
responses are fed back through sensory
channels.
Does this mean that the perceptron iscapable, without further modificationin principle, of such higher order functions as are involved in human speech,
communication, and thinking? Actually, the limit of the perceptron'scapabilities seems to lie in the area ofrelative judgment, and the abstractionof relationships. In its "symbolic behavior," the perceptron shows somestriking similarities to Goldstein's
brain-damaged patients (5). Responses to definite, concrete stimuli
can be learned, even when the proper
response calls for the recognition of anumber of simultaneous qualifying
conditions (such as naming the color
if the stimulus is on the left, the shape
if it is on the right). As soon as the
response calls for the recognition of a
relationship between stimuli (such as"Name the object left of the square."or "Indicate the pattern that appeared
before the circle."), however, theTHE PESCEPTRON
problem generally becomes excessively
difficult for the perceptron. Statistical separability alone does notprovide a sufficient basis for higher
order abstraction. Some system,
more advanced in principle than theperceptron, seems to be required at
this point.
CONCLUSIONS AND EVALUATION
The main conclusions of the theoretical study of the perceptron can besummarized as follows:405

which has not been seen before will becorrectly recognized and associated toits appropriate class (the probabilityof correct generalization) approaches
the same asymptote as the probabilityof a correct response to a previously
reinforced stimulus. This asymptote
will be better than chance if the inequality Pci2 < Pa < Pen is met, forthe stimulus classes in question.
6. The performance of the system
can be improved by the use of a contour-sensitive projection area, and by
the use of a binary response system,
in which each response, or "bit,"corresponds to some independent feature or attribute of the stimulus.
7. Trial-and-error learning is possible in bivalent reinforcement systems.
8. Temporal organizations of both
stimulus patterns and responses canbe learned by a system which uses
only an extension of the original principles of statistical separability, without introducing any major complications in the organization of thesystem.
9. The memory of the perceptron
is distributed, in the sense that any
association may make use of a large
proportion of the cells in the system,and the removal of a portion of theassociation system would not have anappreciable effect on the performanceof any one discrimination or association, but would begin to show up as a
general deficit in all learned associations.
10. Simple cognitive sets, selective
recall, and spontaneous recognitionof the classes present in a given environment are possible. The recognition of relationships in space and time,
however, seems to represent a limit to
the perceptron's ability to form cognitive abstractions.

1. In an environment of random
stimuli, a system consisting of randomly connected units, subject tothe parametric constraints discussed
above, can learn to associate specificresponses to specific stimuli. Even if
many stimuli are associated to each
response, they can still be recognizedwith a better-than-chance probability,
although they may resemble one another closely and may activate manyof the same sensory inputs to thesystem.
2. In such an "ideal environment,"
the probability of a correct response
diminishes towards its original random level as the number of stimuli
learned increases.
3. In such an environment, no basis
for generalization exists.
4. In a "differentiated environment," where each response is associated to a distinct class of mutuallycorrelated, or "similar" stimuli, theprobability that a learned associationof some specific stimulus will be correctly retained typically approaches abetter-than-chance asymptote as thenumber of stimuli learned by thesystem increases. This asymptote
can be made arbitrarily close to unity
by increasing the number of association cells in the system.
Psychologists, and learning theorists
5. In the differentiated environ- in particular, may now ask: "Whatment, the probability that a stimulus has the present theory accomplished,406

F. ROSENBLATT

beyond what has already been done in
the quantitative theories of Hull,
Bush and Mosteller, etc., or physiological theories such as Hebb's?" Thepresent theory is still too primitive, ofcourse, to be considered as a fullfledged rival of existing theories ofhuman learning. Nonetheless, as afirst approximation, its chief accomplishment might be stated as follows:
For a given mode of organization(a, /3, or 7; S or n; monovalent orbivalent) the fundamental phenomenaof learning, perceptual discrimination,and generalization can be predicted entirely from six basic physical parameters, namely:
x: the number of excitatory connections per A-unit,
y: the number of inhibitory connections per A-unit,
6: the expected threshold of an Aunit,
co: the proportion of R-units towhich an A-unit is connected,
NA: the number of A-units in thesystem, andNR-. the number of R-units in thesystem.
Ns (the number of sensory units) becomes important if it is very small.
It is assumed that the system begins
with all units in a uniform state ofvalue; otherwise the initial value distribution would also be required.
Each of the above parameters is a clearly
defined physical variable, which is measurable in its own right, independently of
the behavioral and perceptual phenomena which we are trying to predict.
As a direct consequence of its foundation on physical variables, thepresent system goes far beyond existing learning and behavior theories inthree main points: parsimony, verifiability, and explanatory power andgenerality. Let us consider each ofthese points in turn.

1. Parsimony. Essentially all of
the basic variables and laws used inthis system are already present in the
structure of physical and biologicalscience, so that we have found it
necessary to postulate only one hypothetical variable (or construct)which we have called V, the "value"of an association cell; this is a variablewhich must conform to certain functional characteristics which can clearlybe stated, and which is assumed tohave a potentially measurable physicalcorrelate.
2. Verifiability. Previous quantitative learning theories, apparently
without exception, have had one important characteristic in common:
they have all been based on measurements of behavior, in specified situations, using these measurements (after
theoretical manipulation) to predict
behavior in other situations. Sucha procedure, in the last analysis,amounts to a process of curve fitting
and extrapolation, in the hope thatthe constants which describe one setof curves will hold good for othercurves in other situations. While
such extrapolation is not necessarily
circular, in the strict sense, it shares
many of the logical difficulties of circularity, particularly when used as an"explanation" of behavior. Such extrapolation is difficult to justify in a
new situation, and it has been shownthat if the basic constants and parameters are to be derived anew for any
situation in which they break down
empirically (such as change fromwhite rats to humans), then the basic"theory" is essentially irrefutable, justas any successful curve-fitting equation is irrefutable. It has, in fact,
been widely conceded by psychologiststhat there is little point in trying to"disprove" any of the major learningtheories in use today, since by extension, or a change in parameters, they

THE PERCEPTRON
have all proved capable of adapting
to any specific empirical data. Thisis epitomized in the increasingly common attitude that a choice of theoretical model is mostly a matter of
personal aesthetic preference or prejudice, each scientist being entitled toa favorite model of his own. In considering this approach, one is reminded
of a remark attributed to Kistiakowsky, that "given seven parameters, I
could fit an elephant." This is clearly
not the case with a system in which
the independent variables, or parameters, can be measured independentlyof the predicted behavior. In such asystem, it is not possible to "force"a fit to empirical data, if the parameters in current use should lead to
improper results. In the current
theory, a failure to fit a curve in a newsituation would be a clear indicationthat either the theory or the empiricalmeasurements are wrong. Consequently, if such a theory does hold up
for repeated tests, we can be considerably more confident of its validity andof its generality than in the case of atheory which must be hand-tailored
to meet each situation.
3. Explanatory power and generality.
The present theory, being derived
from basic physical variables, is not
specific to any one organism or learning situation. It can be generalized
in principle to cover any form of behavior in any system for which thephysical parameters are known. A
theory of learning, constructed onthese foundations, should be considerably more powerful than any whichhas previously been proposed. It
would not only tell us what behavior
might occur in any known organism,
but would permit the synthesis ofbehaving systems, to meet special
requirements. Other learning theories tend to become increasingly
qualitative as they are generalized.

407

Thus a set of equations describing theeffects of reward on T-maze learning
in a white rat reduces simply to astatement that rewarded behavior
tends to occur with increasing probability, when we attempt to generalize
it from any species and any situation.
The theory which has been presented
here loses none of its precision throughgenerality.
The theory proposed by Donald
Hebb (7) attempts to avoid these
difficulties of behavior-based models
by showing how psychological functioning might be derived from neurophysiological theory. In his attempt
to achieve this, Hebb's philosophy of
approach seems close to our own, andhis work has been a source of inspiration for much of what has been proposed here. Hebb, however, has
never actually achieved a model bywhich behavior (or any psychologicaldata) can be predicted from the physiological system. His physiology ismore a suggestion as to the sort oforganic substrate which might underlie behavior, and an attempt to show
the plausibility of a bridge between
biophysics and psychology.
The present theory represents thefirst actual completion of such abridge. Through the use of theequations in the preceding sections,
it is possible to predict learning curves
from neurological variables, and likewise, to predict neurological variablesfrom learning curves. How well thisbridge stands up to repeated crossingsremains to be seen. In the meantime,
the theory reported here clearly demonstrates the feasibility and fruitfulness of a quantitative statistical approach to the organization of cognitive
systems. By the study of systems
such as the perceptron, it is hopedthat those fundamental laws of organization which are common to all
information handling systems, ma-

F. ROSENBLATT

408

chines and men included, may eventually be understood.
REFERENCES
1. ASHBY, W. R. Design for a brain. New
York: Wiley, 1952.
2. CULBERTSON, J. T. Consciousness and behavior. Dubuque, Iowa: Wm. C.
Brown, 1950.
3. CULBERTSON, J. T. Some uneconomical
robots. In C. E. Shannon & J. McCarthy (Eds.), Automata studies.
Princeton: Princeton Univer. Press,
1956. Pp. 99-116.
4. ECCLES, J. C. The neurophysiologicalbasis of mind. Oxford: Clarendon,
1953.
5. GOLDSTEIN, K. Human nature in thelight of psychopathology. Cambridge:
Harvard Univer. Press, 1940.
6. HAYEK, F. A. The sensory order. Chicago: Univer. Chicago Press, 1952.
7. HEBB, D. O. The organization of behavior. New York: Wiley, 1949.
8. KLEENE, S. C. Representation of eventsin nerve nets and finite automata. In
C. E. Shannon & J. McCarthy (Eds.),
Automata studies. Princeton: Princeton Univer. Press, 1956. Pp. 3-41.
9. KOHLER, W. Relational determinationin perception. In L. A. Jeffress (Ed.),
Cerebral mechanisms in behavior. New
York: Wiley, 1951. Pp. 200-243.
10. McCuLLOCH, W. S. Why the mind is in
the head. In L. A. Jeffress (Ed.),
Cerebral mechanisms in behavior. New
York: Wiley, 1951. Pp. 42-111.

11. MCCULLOCH, W. S., & PITTS, W. A
logical calculus of the ideas immanentin nervous activity. Butt. math. Biophysics, 1943, S, 115-133.
12. MILNER, P. M. The cell assembly:
Mark II. Psychol. Rev., 1957,64,242252.
13. MINSKY, M. L. Some universal elements
for finite automata. In C. E. Shannon& J. McCarthy (Eds.), Automata
studies. Princeton: Princeton Univer.
Press, 1956. Pp. 117-128.
14. RASHEVSKY, N. Mathematical biophysics.
Chicago: Univer. Chicago Press, 1938.
15. ROSENBLATT, F. The perceptron: A
theory of statistical separability incognitive systems. Buffalo: Cornell
Aeronautical Laboratory, Inc. Rep.
No. VG-1196-G-1, 1958.
16. UTTLEY, A. M. Conditional probability
machines and conditioned reflexes.
In C. E. Shannon & J. McCarthy(Eds.), Automata studies. Princeton:
Princeton Univer. Press, 1956. Pp.
253-275.
17. VON NEUMANN, J. The general andlogical theory of automata. In L. A.
Jeffress (Ed.), Cerebral mechanisms inbehavior. New York: Wiley, 1951.
Pp. 1-41.
18. VON NEUMANN, J. Probabilistic logics
and the synthesis of reliable organisms
from unreliable components. In C. E.
Shannon & J. McCarthy (Eds.),
Automata studies. Princeton: Princeton Univer. Press, 1956. Pp. 43-98.
(Received April 23, 1958)