c.:> r-1 SOME MECHANISMS FOR A THEORY p.4 ' d .J P i .

c : OF THE RETICULAR FORMATION

r' '"i . v<

" 4* I l!

by W. L. Kilmer

W. S. McCulloch J. Blum

(TH u) 9

Division of Engineering Research Michigan State University

East Lansing, ichigan February $1 967

DfstriSu!lon c f this documcnt is unlimited.

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SOME MECHANISMS FOR A THEORY

OF THE RETICULAR FORMATION

3 by W. L. Kilmer, ' W. S. McCulloch, and J. Blum

The ma jo r portion of this work was supported by the AFOSR, administered through Michigan State University. a l so given by the following: Ai r F o r c e Office of Scientific Research Grant AF-AFOSR-1023-66 through Michigan State Un ive r s i Division nd Sjpace Administration, Contract Institute n Laboratory; the National Institutes of Health Grant NB-04985-03 through the Massachu- se t t s Institute of Technology; U. S. Air Force (Research and Technology Division) Contract AF33(615)-1747 through the Massachusetts Institute of Technology; and the Teagle Foundation Inc. , through the Massachusetts Institute of Technology.

Support was

sored by the Bioscience

the Mas s achus ett s

'Associate Professor , D e p a r a e n t of Electr ical Engineering, Michigan State University, and Consultant, Massachusetts Institue of Technology Instrument a t ion Labor a t o r y

Research Associate, Research Laboratory of Electronics, Massachusetts Institute of Technology, and Consultant, Massachusetts Instrumentation Laboratory

Massachuset ts Institute of Technology Instrumentation Laboratory

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TABLE O F CONTENTS

Page

1.

2.

3.

4.

5 .

6.

7.

8.

Appendix 1.

Appendix 2.

Appendix 3.

Appendix 4.

Appendix 5 .

Appendix 6.

Appendix 7.

Appendix 8.

Appendix 9.

Appendix 10.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . Neurophysiology and Neuroanatomy of the R F . . .

1

8

Logical Requirements of the Theory . . . . . . . . . 15

The P r e s e n t Model. . . . . . . . . . . . . . . . . . . . . 18

Simulation Results . . . . . . . . . . . . . . . . . . . . . 35

Form-Funct ion Relations f o r S-RETIC . . . . . . . 39

Conclusions and Future Work. . . . . . . . . . . . . . 47

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . 51

Convergence on a Model . . . . . . . . . . . . . . . . . A l - 1

yk Function Table . . . . . . . . . . . . . . . . . . . . . .A2-1

. t o . Mi Connection Table . . . . . . . . . . . . . . A4-1

Exemplary{yi} . 5' Table (6 M.) . . . . . . . . . . A5-1

Some u . F, C, y. Relationships fo r E . . . . . . . . A3-1 i' 1

th 1

The Prepara t ion Scheme fo r Appendix 5 . . . . . . . A6-1

M. . t o . M. Connection Table . . . . . . . . . . . . . A7-1

Distribution of the Ii - j l in the Mi - t o - M. 1 J

Connection Table J . . . . . . . . . . . . . . . . . . . . . . A8-1

Simulation Results . . . . . . . . . . . . . . . . . . . . . A9-1

Flow Char t of P r o g r a m . . . . . . . . . . . . . . . . . .A10-1

ABSTRACT

Throughout the l ife of the ver tebrates , the core of the central nervous

system, sometimes called the reticular formation, has retained the power to

commit the whole animal to one mode of behavior ra ther than another. Its

anatomy, or wiring diagram, is fa i r ly well known, but to date no theory of

its circui t action has been proposed that could possibly account for its known

performance.

but shallow in computation everywhere, and connected not mere ly f r o m module

to adjacent module, but by long jumpers between d is tan tmodules .

of i ts c i rcui t actions heretofore proposed in t e r m s of finite automata o r coupled

nonlinear oscil lators has failed.

Its basic s t ruc ture is that of a s t r ing of s imi la r modules, wide

Analysis

We propose a radical se t of nonlinear, probabilistic hybrid computer

concepts as guidelines for specifying the operational schemata of the above

modules. Using the smal les t numbers and grea tes t simplifications possible,

we a r r i v e at a re t icular formation concept consisting of 1 2 anastomatically-

coupled modules stacked in columnar a r ray . A simulation t e s t of its behavior

shows that despite its 800-line complexity, it s t i l l behaves as an integral unit,

rolling over f rom stable mode to stable mode according to abductive logical

principles, and as directed by its succession of input 60-tuples.

Our concept employs the following design s t ra tegies: modular focusing

of input inforiliation; modular decoupling under input changes; modular r e -

dundancy of potential command (modules having the most information have the

m o s t authority); and recrui tment and inhibition around reverberatory loops.

P resen t ly we a r e augmenting these s t ra tegies to enable our model to condition,

habituate, generalize, discriminate, predict, and gene ral ly follow a changing

environment.

Our program is epistemological. We are trying to develop ret icular

formation concepts which a r e complex, precise , and valid enough to inspire

reasonable experiments on the functional organization of this progenitor of all

ve r t eb ra t e central nervous t issues .

SOME MECHANISMS FOR A THEORY O F THE RETICULAR FORMATION

1. INTRODUCTION

Throughout the ver tebrate phylum, the ret icular formation (RF) i s

the nervous center which does most to integrate the complex of sensory-

motor and autonomic-nervous signals, thereby permitt ing organisms to

function a s units instead of m e r e collections of organs.

generally of the nervous core of the spinal cord, with bulges in higher

animals in the lower spinal ( lumbar) region, and in regions corresponding

to the neck and brain s t em a r e a s of man ( see F igure 1). In the highest

ver tebra tes it comprises about 1/1000th of the central nervous system.

The R F receives relatively unrefined information f rom all sensory-motor

sys tems which link the organism to i ts environment (visual, auditory,

vest ibular , etc. ) and f rom all internal housekeeping systems which insure

the o rgani sm' s inte rnal well -being (visceral , c a r diovas cular , r e s piratory , etc. ). Its p r imary task, somewhat oversimplified, i s to commit the

organism to ei ther one o r another of 20 o r so, but l e s s than 30, g ros s

modes of behavior - - e. g . , run, fight, sleep, vomit - - a s a function of the

nerve impulses that have played in upon it during the last fraction of a

second.

nerve centers so that they in turn can behave in an integrated, coherent

manner .

The R F consists

It a lso sends out control directives to all other m o r e specialized

In higher ver tebra tes , many variations on the central modal themes

of behavior a r e mediated through a profusion of other bra in regions.

R F in te rac ts with the highest of these regions pr imar i ly through the thalamus,

the information anteroom of the higher brain, by dictating what kind of

functions the ce reb rum is to compute on its input sensory, autonomic,

and mnemonic information. We call this slfunction setting.

tunes f i l t e r s in sensory input pathways to rough-focus the organism's overall

attention; i t modulates motor output signals; it se t s ze ro points in reflex

s e r v o and homeostatic feedback loops; it controls the organism's sleep-

waking cycle and postural substratum; it participates with the hypothalamus

i n the regulation of vegetative activities; etc.

The

The R F also

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CRANIAL PORTION OF THE RETICULAR FORMATION

CEREBELLUM

BRAIN STEM

CORE O F THE RETICULAR

NEURAXIS \\\

LUMBAR PORTION OF THE RETICULAR FORMATION

$ E l a Q-lc, o r n

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But the R F does not do everything. F o r we note that though 8 decerebrate adult cats can distinguish between tones, they cannot between

tunes; and though they can see brightness and respond appropriately to

simple moving forms, they apparently a r e devoid of all refined visual

perception. Neither do t h e y orient well t o bodily touch, cold, p ressure ,

o r shock stimuli.

stereotyped; they a r e modal. They lie, sit, stand, walk, run, fight,

sur render , sleep, eat, drink, vomit, defecate, micturate, and mate.

Their stalking, pouncing, directed cuffing, skillful playing, and other

such activities, a r e essentially gone.

capacity for long-range intentions o r complex problem-solving.

they have left of the i r nervous sys tem is too busy trying to keep them

alive f o r that.

important stimulus contingencies, because they must in the i r natural

domain retain a s much of the i r quick response capability a s possible,

and it takes all the neural c i rcu i t ry they have left to do it.

Their movements a r e impoverished and highly

They a r e stimulus bound with no

What

They a r e able only to cope with the most urgent and

If the behavior of decerebrate cats means what we think, the

R F in its natural milieu is certainly not a refined, precise, art iculate,

o r temporally-sophistlcated mechanism by whole-brain standards.

is f a r m o r e integrative than analytical, and far m o r e comprehensive

than apprehens ive. 34 Yet one must not delude himself on this head.

F o r even cockroach ganglia, consisting of only about 1000 primitive

neurons, can adaptively s o r t out 29 f r o m 30 pulse signal burs t s

and the R F is by all measu res a wizard compared to one cockroach

ganglion.

It

45

P. has noted that R F s in experimental cats always

s e e m t o have a preferent ia l set point instead of constantly and gradually

passing f r o m level to level. P. Dell24 has discussed the functional

stabil i ty of R F s in t e r m s of their homeostatic tendency to eliminate

input disturbances throagh correct ive effector actions and by resett ing

input-filter operating points.

neu ra l load adjustments, such as bulbar R F elevation of a r t e r i a l p re s -

s u r e through neuPohumoral secretion,or reticulo-cortico-reticular

depress ion of neural activity through feedback regulation.

Sometimes the correct ive actions a r e

A t other

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t imes the correct ive actions involve CO

neural activity in both ongoing and recrui table (in eme rgencies) resp i ra tory

centers , o r involve selective inhibition of some groups of interneurons

(intermediate diagastr ic jaw -opening neurons, for example) with com-

pensating facilitation of others (masse t e r jaw-closing motoneurons, for

example).

fractionation of reflex actions.

afferent pathways to a neural center and switch in another.

Dell 's homeostatic observations fur ther cor robora te our mode concept

of R F function.

o r 0 2 2 induced changes in R F

The la t ter often causes a dras t ic alteration of the cort ical

It can a l so switch out one group o r

Wall and

Let us t r y t o develop our notion of what the R F does in another

way. A t the millipede s ta te of evolution, the R F is essent ia l ly the ent i re

cent ra l nervous system. By the pigeon stage, it has grown, o r separated

out, s eve ra l comparatively specialized computers for making finer

discriminations between sensory stimuli and for computing m o r e prec ise

motor control signals, than it could possibly produce by itself and s t i l l

maintain its fast-acting overall command and control function.

among these specialized computers a r e the visual, vestibular, bodily-

sensory, and auditory systems, and the cerebel lum to compute prec ise

auto-correlations for actions of the pigeon on the pigeon and the pigeon

on its world (as required for pecking, control in flight, etc. ). The

pigeon R F has a lso evolved specialized basa l ganglionic mechanisms

for programming its associated bodily movements ( required for running,

fighting, feeding, mating, etc. ); a s e t of well localized feedback paths,

called simple reflexes; and a set of regenerative nerve loops for con-

troll ing various types of internal rhythms (cardiovascular, respiratory,

digestive, etc. ). But it i s still c lear , especial ly f r o m the work of

Lorenz and his school, that the pigeon, s e a gull, goose, and other

organisms of that evolutionary rank behave in a nea r ly modal fashion.

Chief

By the humanstage of evolution, the R F has grown a cor t ica l

mantle over the r e s t of its phylogenetically older s t ruc tures .

older s t ructures , when left by themselves, a r e only concerned with the

ra ther m o r e immediate preservat ion of the individual and its species.

But in humans, with their additional cortex, we find new and different

These

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types of functions, like language; we a l so find that the behavioral influences

of many of the older functions, like anger, a r e great ly modified, and that

an enlarged frontal lobe has mushroomed the development of long-range

judgement, sophisticated attitudes, and deliberative purposes in the

organism.

and motor outflow computers a r e l a rge r and m o r e intricate than ever . But,

for all the R F ' s reliance on the discriminatory, associational, memory,

abstractional, computing, and programming powers of the cortex, it

apparently has never relinquished its cen t ra l command function to the

cortex. Only

the R F has a wealth of direct o r monosynaptic connections to and f r o m al l

other cent ra l nervous s t ructures . 81y 8 9 Only the R F is able to arouse,

put t o sleep, and turn off (over-ride in a c r i s i s ) the r e s t of the ent i re

forebrain. 7 0 And only the R F has the position and connectivity to possibly

make computations wide enough (of sufficient scope) and shallow enough

(in logical depth) to always a r r i v e at good g ross modal (integrative)

decision within a fraction of a second, given the requisite information.

We find, too, that the visual, auditory, bodily-sensory, 116

The evidence for this is both anatomical and physiological.

89

Our modal command concept of the R F is not incompatible with ak

the amodal behavior of orches t ra pianists, o r active and learning ca ts

that have almost completely ablated RFs , o r incinerated martyr pr ies t s ,

o r such things a s the m e n of New Hebrides jumping head-first off 50-foot

towers only to be stopped 3 inches f rom the ground by thongs tied around

the i r ankles.

development.

ex t r eme cort ical "control" of the R F . demonstrate the neurological dictum that brains concentrate first, with

whatever equipment they might have left, on staying alive. In this

sense, no bra in region is totally sacrasanct with respect to any major

b ra in function.

modulated behavior can get. With this, let us delve a little m o r e

deeply into the relations between brain regions that these examples imply.

It is t rue these things pose some problems for our

The mar ty r -p r i e s t s and tower jumpers operate under

The violated cats dramatical ly

The pianists demonstrate j u s t how nonmodal cortically

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"This paragraph is most ly a rebuttal to comments made by engineers on an ea r l i e r vers ion of this report.

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J u s t a s the commander of a fleet might have to plunge his ships

into destruction in o rde r to achieve a mission formulated by his miss ion

control office, the R F might a l so have to a c t analogously in relation to

some specialized brain agency it begot.

to other bra in centers j u s t a s a fleet commander m u s t t ru s t and l isten to

the fleet 's radar , gunnery, engine room, combat, and navigational offices.

After a modal command has been issued, both animal and fleet must c a r r y

it out according to a preprogrammed se t of rules, with embellishments as

contingencies demand. In the mar ty r -p r i e s t and tower-jumping examples,

the R F ' s command to the r e s t of the central nervous sys t em is the best

reconciliation of a host of conflicting demands placed upon it.

hand, piano playing is a cortically-mediated activity requiring the fractionation

of m o r e primitive response patterns into a special blend of prec ise actions.

The R F permits this kind of thing, but does not command o r control it

(except by default). In the cat case, we see that animals can survive

massive R F lesions, animate, and even condition a f te r them. This is

because brain functions can migrate , given the necessity, t ime, and a

decent anatomical chance, in o rde r t o get the i r p r imary jobs done.

larly, if the conductor of an orches t ra pas ses out, a front row instrumentalist

can take charge, but usually only a t considerable cost to the o rches t r a ' s

quality of performance.

F o r the R F must t ru s t and l isten

On the other

Simi-

The R F is thus "general" - - or in c lass ica l Greek terminology, Its relation to other bra in s t ruc tures reca l l s "first , I t -.- in the brain.

the Biblical passage, I ' . . .bu t whoever would be grea t among you m u s t

be your servant, and whoever would be f i r s t among you m u s t be your

s lave. ' I

Now consider the R F , minus everything on its input side (generally

speaking the dorsolateral regions). i ts output side (ventral- la teral R F ,

basal ganglia, etc. ) ? a l l of its local ref lexes , and a l l of its r e sp i r a to ry

and other rhythmic operational aspects which a r e functionally separable

f r o m its main decisionary tasks. Denote what is left R F . 'k The task of

RFXC we take to be a blend of modal-commitment and function-setting

activit ies; and the la t ter we take to be engendered by the former .

is the fundamental assumption ---- of o u r I paper , validity a t the medullary l e v e l of the RF.

This

It has the g rea t e s t apparent

More rostral ly , at the midbrain

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and thalamic levels, the R F decisionary functions grade off into predomi-

nantly nonspecific (function - setting) and a s s oc iat ional - integrative activities . Let u s denote by R F * a sharply modal idealization of R F * ’ s functions.

will be the object of our study f rom he re on,

leas t a promising theoretical progenitor of a real is t ic R F * concept.

R F *

and we will regard it a s a t

We will next sketch the known R F neurophysiology and neuroanatomy,

and then go on to propose a theoretical framework for R F * that we think

stands a chance of being right enough and developmentally promising enough

to eventually be of some use t o us in understanding real R F s .

a r e a f te r f i r s t is a way to think about how R F * always a r r i v e s a t an integrated

modal decision in a dozen or s o neural decision t imes instead of disagreeing

among its severa l par t s in the face of competitive o r contradictory input

signals.

indispensable clue t o how this is done instead of an i rksome o r gratuitous

constraint. Some would not ag ree with us. They must then advance along

the l ines of categorical philosophy or psychology, which is not our p r imary

interest . We want to know, a f te r Clerk Maxwell, not only the go of our

mechanism, but the par t icu lar go of it.

What we

We believe that the highly character is t ic R F anatomy is an

The magnitude of our problem is signaled by some previous

theoret ical results.

( l inear a r r a y s of finitely but unboundedly many identical descre te automata

information-coupled in both directions) have probability 1 of not being able,

a f t e r start ing in equilibrium, to a r r ive a t an equilibrium point in l e s s than

a bounded number of component-automaton decision t imes following a 49, 50 perturbation of inputs.

Other resu l t s on i terated logic nets 41’ 47’ 48 point up related

We know that logic nets organized along R F lines

difficulties in our problem.

cer t i fy that coupled nonlinear oscillator manifolds would put u s in any

be t te r s tead on such accounts.

difficulty hinges on the fact that we a r e concerned with t ransient o r

decis ionary processes by which complex nonlinear decision-making

s y s t e m s rol l over f r o m stable mode to stable mode, and not the steady

state effects in such sys tems (cf. Appendix 1 ) .

We recal l the complete lack of methods to

It must be emphasized that our central

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2. NEUROPHYSIOLOGY AND NEUROANATOMY OF THE R F

The Scheibels have so f a r done what for u s is the mos t definitive 93 neuroanatomy available on the R F . In the i r milestone paper of 1958,

they car lcatured the ar,&tomkal s t ruc ture of the lower 2/3 of the R F in

the bra in s tem by comparing it to a s tack of poker chips.

region the dendritic processes of R F neurons ramify in the plane of the

chip face, often covering near ly half of the face a rea .

Nauta

neither tufted nor wavey, but consisting of long shaftlike processes whose

branches a r e usually longer than the s t e m of their origin.

of nearby neuroqs, there is a ve ry large degree of overlap and intermingling,

as shown schematically for the bra in s t e m region in Figure 2.

v e r y s imi la r to Scheibels' Figure 1 in Reference 86. )

In each chip

The Scheibels and 81 describe the shape of R F neural dendrite a r b o r s a s primitive,

Among dendrites

(This i s

The dendritic organization of the nerve nuclei that furnish inputs

t o the R F is predominantly longitudinal, as seen in Figure 2b, or of a

tufted or wavey character . The axons out of these nuclei, and the axonal

collateralizations out of a l l of the longitudinal f ibre t r ac t s that feed into

the RF, turn off sharply to reach into the R F in the planes of the R F ' s

grea tes t dendritic ramification.

dozen or more different input systems m a y synapse on a single R F neuron,

and each R F input nucleus and f ibre t r a c t in genera l feeds ve ry many R F

cross-sect ional levels, the Scheibels suggest that the R F might tolerate

considerable puddling of information a t each of its c r o s s - sectional levels,

but demand somewhat grea te r informational r igor between levels. Nauta

s imilar ly regards the R F dendritic anatomy a s an isodendrit ic ma t r ix

which serves a s the s t ruc tura l substrate f o r a near-continuum of signals.

Since in this process a s many as a half

The order of magnitLtde of the number of R F afferents, the number

of R F neurons, and the number of R F efferents , i s accepted as about the

s a m e f r o m frog to man.

60° ventrally f r o m more ventral R F ce l l bodies, and about 180

la teral ly f r o m m o r e dorsa l R F cell bodies.

situated R F neurons a r e in general sma l l e r than the i r m o r e medial

counterparts. l 1 7s l 1

with local operations, and l a rge r ones, with global functions.

R F dendrites generally appear t o fan out about 0 dorso-

All p rocesses of m o r e la te ra l ly

SmalLer R F neurons a re , of course, m o r e concerned

In general ,

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a) R F CROSS SECTION. SHOWING DENDRlTlC ORQANlZATlON ON LEFT. A N D INPUT ORGANIZATION 0 N . R I O U T .

HEAD END TWELTH NERVE FOOT €YO NUCLEUS (TONGUE) c

SPINAL

NUCLEUS (CEREBELLAR, ETC.1

b) RF LONGITUDlNAL SECTlON SUOWINQ DENDRITIC RAMlFlCATlON FROM NEURAL CELL BODlES. AND ORGANIZATION OF INPUT NUCLEI. C) POKER CWIP CARICATURE OF a) A N 0

Fig. 2. Brain etem RF dendritic anatomy.

9

m o r e ventral R F nemons participate m o r e extensively in effector

functions, more dorsa l R F neurons in sensory functions, and m o r e

l a t e ra l R F neurons in vegetative functions.

The R F axonal anatomy corresponding to Figure 2b is shown in

F igure 3. (This drawing is essentially Scheibels' Figure 4 of Ref. 86. ) In Figure 3 a character is t ic R F axonal process is seen coursing i ts way

longitudinally over a ma jo r portion of the brain stem.

off into other R F levels and var ious R F input nuclei, a s well a s into both

corticifugal (i. e . , descending) and corticipetal (i. e . , ascending) neural

f ibre t rac ts .

Collaterals branch

A good many R F axons a l so project nonspecifically'k into - ~.

ce reb ra l regions, 72-81 as well a s directly out to the level of the first

neuron in each of the sensory sys tems (e. g., to the retina of the eye,

and to the muscles of the Inner e a r ) . 43 In short, the R F sits athwart all

incoming and outgoing Ctrvous transactions ca r r i ed out over the ent i re

neuraxis, and it both samples and modulates the i r spatio-temporal infor-

mation sequences s o a s t o command the g r o s s modal operation of the

organism.

Essentially a l l that is known about the Nissel archi tecture of the

R F is that there is a full range of neural cel l sizes.

c i rc imscript ion of neural groups nor laminar nor other striking distributive

organization is evident. This i s quite opposite of those regions just outside 81 the R F core (Nauta's definition of R F

But neither neat

). Thus far no very helpful hypotheses have been developed relating

the prec ise geometric forms and s izes of R F neurons to the type of functions

they compute. In this sense, the R F is much m o r e intractable than the 56 f rog 's eye.

Near any given R F neural cel l body the re m a y be tens of thousands

of both fast conducting (1 00 me te r s / sec ) insulated f ibres and slow conducting

(a few meters / sez) uninsulated f ibres , but the functional significance of

this has only been guessed at. We know that R F neurons character is t ical ly

,k "Nonspecific" projection te l l s cor tex what functions to compute, but does not furnish the information for computation. t o most ly passes through at least one thalamic-level (i. e., diencephalic) synapse.

The projection r e fe r r ed

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23, 24 respond to exceptionally wide ranges of neural lo’ 97 and chemical

st imuli involving perhaps severa l sensory and vegetative modalities.

example, in cat there a r e R F neurons that increase the i r firing r a t e s

during asphyxia, tickling of ha i r cel ls in the nose, and postural unbalance.

Other R F neurons respond to v j sce ra l disorders , crude body interface

phenomena (touch, pressure , and cold), and cer ta in phases of anti-gravity

bodily kinetics; and s t l l l others t o ce reb ra l control signals, raw information

f r o m head-end distance receptors (eye and ea r ) , and signals f r o m neuro-

endocrine receptors In the hypothalamus.

F o r

We know a lso that there i s mass ive ret icular involvement in

motor outflow and attention-focusing affairs. F o r example, r a t s do not

ordinarily distinguish yellow ; but if they a r e hungry and small cheese,

R F directed outflow sets up visual computations which enable them to.

Also , Hernandez-Peon4’ has found cochlear nucleus (auditory pathway)

neurons in anesthetized cat that respond well t o clicks until odorous fish

i s placed under the ca ts ’ nose. Thus a modal decision might well be viewed

as a v e r y broad command to attend, e. g., t o running, feeding, o r fighting.

P. Dell has developed this notion in t e r m s of c r i t i ca l (in the sense of

j ud g eme nt ) reactivity. 24

In the plastic domain, R F neurons a r e the first to adapt out the i r

responses to intense but meaningless st imuli (e. g., gunfire at a shooting

range), and a r e generally the f i r s t to show signs of conditioning to sensory

indications of imminent painful stimuli. 39’ 59 But the R F does not l ea rn

ve ry much on a long t e r m basis. P. Shurrager ’s dog whose spinal cord

had been t ransected as a pup, learned to walk, sit, lie, copulate, and void

urine and excrement with near ly the right pos tures and motions. A soldier

will sometimes hit the d i r t on hearing any loud, s h a r p sound for years a f t e r

a long battle experience.

usually even retain conditioning for m o r e than about 30 minutes.

But these are ex t reme cases . The R F does not

There is strong evidence to suggest that the R F can change modal

commitments a t a steady ra te of not m o r e than about 3 t imes pe r second

(spinal reflexes, some of whose paths can be t r ave r sed in about 20

milliseconds, notwithstanding), but m u s t be dr iven with pulse repetition

ra tes of the ordcr of a few o r a few hundred pe r second f o r this t o occur. 124

12

-

I i

i I

I I I I I I I I

Most pulse r a t e s at low power have very l i t t le overall effect inside R F

t issue.

probably the focusing down affects following the c r e s t of this activity

pe r s i s t for a minute o r more .

( i s t he re a lion behind that bush o r not?) a r e produced at the r a t e of about

ten pe r secondo and that this i s the main limiting temporal factor in those

cort ico-ret icular exchanges that pr imari ly concern modal decisions. We

note that humoral and hormonal rhythms, with periods of hours, days,

months, and yea r s , can have a grea t effect on the overall sensitivity and

s e t of a RF.

If the R F i s engaged in a significant overall decisionary activity,

5 W e conjecture that cor t ical perceptions

We understand something of how vertebrate organisms take habits

and conduct their neuronal affairs at the ret icular level. It has remained

the same f rom shark and lamprey to man.

nothing about the kinds of spatio-temporal information codes that R F employs

to cope with i t s horrendous intraret icular communication and computation

problem. Granted, there i s some signal channeling a s a function of pulse

repeti t ion rate that has often been noted experimentally. 3 2 And a good

deal of recent spike interval histogram119 and noise power work120 has

re inforced the old belief that much neural processing must be statist ical .

Also Lettvin, McCulloch, and co-workers have demonstrated how single

neurons could compute any logical function of their inputs, indicating

that the action potential 's al l-or-none charac te r i s not accidental.

R F spike waveshapes have been reported to often have very long trail ing

edges (up to 10 ms); neural-glial field effects f igure ever m o r e prom-

inently in R F operation;

continaully outcropping a s possible distributed R F operating conditions.

So all is not combinatory logic, anatomical pathways, and statist ical pulse

frequency processing either. In a different vein, we can neither imagine

how, nor whether, R F input information fractionates into small, coherent,

m o r e o r less loosely coupled Hebbsian assembl ies that intermingle to

r ea l i ze associative, cooperative, and intergrative decisionary behavior.

In the European newt, only the R F can regenerate the r e s t of the

whole brain. Thus in some sense, i t knows what it needs to help it run

the organism in which it res ides .

whereas the r e s t of the forebra in is not.

81 But we know essentially

But

97 2 and R F biophysico -biochemical pa rame te r s a r e

2, 23

In the newborn, the R F is mature

Apparently, give o r take a few

1 3

lesions, the R F is the sine qua non of viability whereas the other brain

regions a re not -- at leas t ear ly in life.

relations with other bra in regions, but disappointingly little on intra-

ret icular ope ration.

In summary, a good deal is known about the R F ' input and output

systems, and something about its neurobiology, Much is lacking on the

detailed connection pat terns between the var ious neura l types, and the i r

counts With changes of position. Prac t ica l ly nothing is known about how

R F inputs s ta r t appropriate computations racing up and down the net S O

a s to always y ie ld effectively unanimous decisions for modal command

signals .

These facts shed light on ret icular

We a r e convinced that i f the R F is ever t o be real ly understood,

we must have a theoretical model that will enable us

logically sophisticated experiments of sufficient cybernetic dimension-

ali ty and complexity to take into account those differences which make

a difference.

of one N-dimensional experiment in a thoroughly N-dimensional system.

to intuit f r o m it

N one-dimensional experiments can never take the place

14

3. LOGICAL REQUIREMENTS O F THE THEORY

First of all, we must understand what kind of logic the RF::'

described in our introduction performs. C. S. P i e rce called it the logic

of relations, but for its c leares t statement we go straight t o the father of

modern biology, Aristotle. He described three kinds of logic: deductive,

inductive, and abductive. RF:k does the latter. Its scheme i s to go f r o m

the facts and a rule to a case, i. e., the facts of sensory and internal

perception as represented over the RF:: input channels, and a rule fo r

classifying these facts by an overall computational scheme; to a c a s e of

the form: the input i s X and this implies action mode Y.

- - -

An organism's R F case s t ruc ture is always the resul t of its

evolutionary, developmental, and experiential heritage.

Computer theoris ts generally think of deductive, inductive, and

abductive logic as follows:

deductive logic 2r

inductive logic 2r

abductive logic -

execution of a given program

generation of a new program during a training

period such that the new program can produce

a proper output given any input closely related

to one in the training set.

selection of the appropriate p rogram f r o m a repertoire in accordance with a rule for

analyzing program requests. Since these

program requests can be made in any f o r m -- e. g., in na tura l language -- in general a

calculus of n-adic intentional relations is

needed fo r the analysis.

After each new modal decision, R F * keys the proper output

program, and f r o m then on as far a s it is concerned everything follows in

a completely deductive manner. (For example, the programmed output of

the basa l ganglia throws its keyed signal sequences for walking down over

the interlocked entrainments of nerve centers in the a r m s and legs, and

they in turn embellish the details of the o rde r s given them as the contin-

gencies of rough terrain, etc., demand; and s o on out to the periphery,

15

where the effector signals a r e t ransformed into smooth and complicated

actions. )

Doubtiess R F * never computes single modal decisions directly,

but ra ther their half-center representations. To i l lustrate, F igure 4

shows the half-center dimensions for the lumbar enlargement of a dog.

The advantage of such a representat ion is that for n dimensions, a

single 2n-valued function can be replaced by n two-(or sometimes m o r e )

valued functions . Conditioning, habitualion, and long-term learning in the R F requi re

only inductive logic in principle, and quite probably a t this stage a r e bes t

studied in cortex.

1 6

m a :imum of forward motion

ra

f a s t dog-paddle swim

------------- ' I

/ ' I ----------

/ /

/ --- t e fast frog- kick swim slowest dog-

should be added alon I

I slowest frog-kick I swim I asymmetr ic

- - - - u r i w o r , - point

ze ro forward s ymme t r i c ~ L4,% and

motion

shake anti- gravity anti - +. lie system out gravity

system in

Figure 4. Half- center dimensions for lumbar R F of dog. (Note: Heavy lines show

dimensions. )

17

4. THE PRESENT MODEL

This section descr ibes our present R F * model, which we denote

S-RETIC. It i s a car ica ture of the poker chip analogy for the bra in stem

R F , and was constructed to be as simple a s possible without violating our

intuitive notion of what R F operation is like.

In the model we replace adjacent groups of the Scheibels' approxi-

mately 1 00-micra-thick poker chip regions by single modules which contain

nonlinear, probabilistic hybrid computers. We require that all modules

in the result ing columnar a r r a y be similar and operate on the same

synchronous t ime scale.

in a way suggested by the known R F axonal anatomy.

the guide in specifying external S-RETIC inputs to each of the modules.

The modules a r e interconnected to a degree and

The anatomy is a l so

The plan is somewhat different on the S-RETIC output side. Since

it is mainly the computational s t ruc ture of single R F modal decisions that

we a r e interested in, each of the S-RETIC modules is given only d i rec t

mode-indicator outputs. We as sume that the effects of each S-RETIC mode

change show up at some la te r t ime over S-RETIC's input lines.

departs f r o m R F biology in that our S-RETIC outputs have no d i rec t way

of influencing the supposed organism's input and output sys t ems which

feed it.

Doty's, 28 which show that even brain stem swallowing motoneurons " seem

to have an unpredictable and random pat tern f r o m one swallow to the next,

though the overall schedule of excitation and inhibition among the par t ic i -

pating muscles in highly constant. I t

This

Such an overall-output approach s e e m s justified by resu l t s like

Figure 5 shows a reduced schematic of S-RETIC. All u and yi

lines a r e binary (an a r b i t r a r y but convenient bas i s of information coding):

the Mi a r e S-RETIC's poker-chip logic modules; the Si correspond to the

var ious humora 1, chemoreceptive, and exte roc ept ive and inte roc e pt ive

sensory and internuncial sys tems that feed inputs direct ly into the R F ;

the Qi(only St the upper and lower boundaries, T and B, correspond roughly to the mid-

bra in and high cerv ica l regions of the higher ver tebra tes , respectively.

F o r c la r i ty each type of connection appears in F igure 5 only once, whereas

actually the connection types proximate to M

shown) a r e the modular mode-indicating output l ines; and 7

r e c u r at all corresponding 7

18 I

“7 “a

simulated input

Figure 5.

s imulation model.

S -RE TIC

- -

asc line a

I I

(k)

I

19

s imi la r locations over the entire figure.

diverges f rom o r converges to one o r a group of S . o r M. in Figure 5, it

does likewise at every corresponding s imi la r location in S-RETIC. Each

M then, receives inputs f rom severa l but not all S and each S. feeds

severa l but not all Mi'

not all other Mi, and receives information direct ly f r o m seve ra l but not

a l l other M

Thus, i f a connection type

J 1

i' j' J Each Mi feeds information direct ly to seve ra l but

j ' The M.-to-M. connections a r e arranged s o that, in general , modules

close together a r e information-coupled more closely than modules far apart .

This is in line with the neuroanatomy. Similar res t r ic t ions a r e a l so all

that govern the te rmina l distributions of the A and C bundles, though the

anatomy of Nauta and Brodal suggests that somewhat m o r e patterning and

specificity than this exis ts in the r ea l R F .

and descending bundle s izes a r e delimited to 5, 4, and 4, respectively;

and the degrees of S.-to-Mi fan-in and fan-out are delimited a s suggested

by the R F input anatomy (involving nuclear regions, f ibre t rac ts , and m o r e

localized la teral re t icular s t ructures) . The prec ise nature of these de-

limitations i s suggested in Figure 5, and is specified in detail in Appendix

4. "use" distributions on the y. and u i .

relations a r e important because the information on these l ines should be

highly but nontrivially correlated, with the degree of correlat ion between each pair of y. determining their proximity in the y1 1, . . . , y42 ordering.

The y. a r e realizations of all 42 symmetr ic switching functions of the f o r m

(u . A u .) V (u . A u ) V (v

i, j, 5 7, and k =- 8 o r 9. 5

This keeps the percentage of 1 ' s on the y . ' s about the same as that on the

u Is (cf. Appendix 3 for details) , and p r e s e r v e s some useful distance

propert ies in the passage f r o m u i 1 connections a r e made randomly s o that the probability of a d i rec t

M. -to M. connection is inversely proportional t o the absolute magnitude

of the square root of (i-j). (Cf. Appendices 7 and 8 f o r connection table

and details. 1 Z and E in Figure 5 a r e thus included only to simulate an

R F environment that engenders input signals f r o m a covarying world.

(For example, i f a runaway c a s stops abruptly at a wal l at the bottom of

1 J

The S . output and Mi ascending J

J

The specification was made with the intention of imposing fair ly even

The corresponding y. connection 1 1

1

1

A u i), with i, j, and k pairwise different, 1 J J k

(Cf. Appendix 2 for tabulation of functions. )

1

i

j to y. signal sets . The Mi-to-M

1 J

a hill, a witness is likely to hear a crash.

then t ransmi t correspondingly covariant signals into his re t icular formation. )

His visual and auditory pathways

Before discussing the details of the M. in Figure 5, we note that R F 1

biology recommends to our use the following M. input design strategy, which 55 not so incidentally i s aligned with Leibnitz's notion of the diversified monad

(especially for { oi}, { yi> , { yi } , and {Mi} se t s izes of over 50, 1000, 50,

and 100, respectively, which i s L o r e l ike what we a r e really thinking of

anyway. We chose the small numbers 9 , 42, 5, 12, because they were

the smal les t we thought wecould get away with without completely violating

our RF::' concept).

inputs which jus t enables it to get a good picture, o r relatively high resolu-

tion view, of the signal s ta te in a cer ta in small portion of the bundle, but

which only permi ts i t a progressively poorer picture of the signal s ta te in

portions of the Z bundle more outlying f r o m its "area centralis ." This

makes each M. both a general is t and a specialist.

general is t , S-RETIC would not be able to discriminate well enough.

each w e r e only a specialist , S-RETIC would not be able to piece together

a good global decision; in addition i f any Mi should fail, the overall sys tem

would go totally blind in some

R F::: s y stem.

1

Each M. of S-RETIC should receive a selection of yi 1

If each were only a 1

If

a r ea , and that should never happen in an

F igure 6 depicts the essentials of our notion. Over a c ross -sec t ion

yi of the

inputs exactly k units of information on the signal s ta tes of all the u l ines

within each marked off a r e a of the cross-sect ion (A o r B, for example).

A r e a A, then, belongs to the F igure 6 M.'s a r e a central is , and B its

per ipheral low -resolution a rea .

bundle, the Mi for which the figure was drawn derives f r o m its

1

The idea i s to have each M.'s a r e a central is displaced f r o m each 1

other M. 's a r e a central is , but such that each point in the {IT) bundle's c r o s s

section is at l e a s t near the a r e a central is of some M Then each Mi

knows something about all of S-RETIC's input affairs , so is diversified,

but is a specialist on only a subset of them. This admits the necessary

1

i '

~

:F "V I ' i s logical inclusive o r , lland''A1l is logical "and, I ' with (1 A 0) = 0

and ( l v 0) = 1.

21

bundle of Z lines into E of Figure 5.

Mi gets k units of information f r o m each outlined area of the bundle's c r o s s section

Figure 6. Basis of our module design philosophy.

22

I I 1 I 1 ll I lI Il

I

I

S-RETIC acuity.

one can shoot holes through S-RETIC, and what's left performs with an

overall decisionary acuity that is roughly proportional to the number of

good Mi it has left. Careful checking will reveal that S-RETIC real izes

the foregoing {u ;> -to-(yi} -to-{Mi} design strategy.

much m o r e justifiable y function specification than the one given h e r e in

our present work on trainable S-RETICs. )

Another important consequence of this s t ra tegy i s that

(We are using a

Everywhere above, the Z bundle does not correspond to any signal

paths in the r ea l world.

relations of like charac te r occurring in r ea l world neurology over R F

input pathways.

But at the {y.} level, there should be signal co r - 1

Figure 7, t o be explained later, shows the scheme we chose for

It was inspired by the des i r e to rea l ize the realizing our Figure 5 M..

logic, if not the mechanisms, of coupled nonlinear oscil lator manifolds.

This was because, to a reasonable approximation, that ' s what the R F is.

The mos t ancient types of neural t issue always suggest this.

1 te l l s why we did not actually use such manifolds in our model. ) But

we could a l so see that we would have to organize our logic along cer ta in

s t ra teg ic lines in o rde r to maintain some measu re of control over our

model 's behavior.

1

(Appendix

We chose two major s t ra tegies in addition to the above a r e a

cent ra l i s M. input strategy: 1

(a) Module recrui tment according to a redundancy of potential

command.

have the mos t information a l so have the most authority. It a l so requiresthat they be able to express the i r authority by

recruit ing other modules with l e s s apparent information

over to the i r modal persuasions.

This requires that those modules that apparently

(b) Module decoupling a t t imes of S-RETIC input change. This

enables modules to a r r ive at relative modal preferences

following each overal l input change most ly on the bas i s of

their own direct y inputs.

with the rest of S-RETIC, interact with each other, and

eventually converge on a single output modal consensus.

They then gradually couple in

23

In a, k 5 M .d cr w 0

sd l-4 cd u a h

.r(

c,

5 M

cr .rl

24

F o r a g ross tempora l picture of how S-RETIC does its computing,

then, we can imagine that S-RETIC has j u s t received a new overal l input.

Each M. first computes its corresponding modal preferences, mos t ly on

the basis of its new y inputs. At the next t ime step, it exchanges modal

preferences with selected other modules; concomitantly, it adjusts its

own preferences to some degree as a function of those jus t received f r o m

other Mi. It continues t o operate in this manner each t ime s tep there-

after, coupling in eve r m o r e tightly to the other M. as it goes along (to

a limit), until the next overal l input change.

a consensus of modules is swung over t o a single modal preference and

is held there until the next overal l input change occurs.

whole sequence starts all over again.

1

1

Somewhere in this process ,

At this point the

Let us now take a coa r se look at how each M. works. Afterwards, 1

we will go back and descr ibe precisely how the foregoing s t ra teg ies a re

implemented to achieve the desired S-RETIC behavior.

M. 's probability computer computes f r o m its five present y. inputs the

p1 vector (pIl, . . . , p14), where p ' is the probability that theJpresent over-

all (vi} signal configuration properly corresponds t o a j th mode output

indication.

as shown in Appendix 4. (Appendix 5 tells how Appendix 4 was derived.

Summarily, Appendix 4 was computed f r o m a random a pr ior i ass ignment

of each overal l {vi} signal configuration to one of our four modal cate-

gories. At present, however, we are trying to train in S-RETIC's

Appendix 4 tables right f r o m scratch, where all p' = . 25, by rein-

f o r c ement procedure s . )

In Figure 7,

1 L

j

This computation is actually only a ta.ble lookup, using tables

j

The p f t i and ptt'i signals into the M. in Figure 7 give the momentary

Each 1

modal probabilities as evaluated by selected M. above and below M.. J 1

N box is a normal izer such that the s u m of Its four analog outputs equals

1. The twelve normalized pli, p'Ii, P ' " ~ values are componentwise

operated on by a nonlinear function, f , as shown in Figure 2.

are computed through C , Cr, C a 6 unit according to the formula

The pi multiplier units and an Av averaging

25

where Cr = C

The pi, then, a r e jus t weighted averages of p1

the i r t rends have been appropriately exaggerated by f.

d iscuss f and the Cs a s mechanisms fo r redundancy of potential command,

and the Cs a s mechanisms f o r decoupling.

C Q, C 6 , and Ca a r e determined as indicated below. =1 =2

p'li, p"' values af ter i' L a t e r we shall

Figure 8 gives the f function.

Since (Gl, p,, p,, F4) i s not in general a probability vector (all

components 2 0, components sum to 11, we put (Fl, p2, p3, p,) through

the h, T, h , N blocks to make it so. These blocks also pe r fo rm the

indispensable function of nonlinearly interacting the four probability

modalities so that S-RETIC is not jus t four single-mode probability

computers in parallel.

unit in U D (for Unit Delay), and used as Mils output to the ascending a

s t r eam, and the overall S-RETIC output bundle.

- - - -1

The output of h, T,h-', N is delayed one t ime

Since all of the Mi do not usually agree with probability 1 on which

mode the overall { yi} signal configuration properly corresponds to, we

specify a general output modal decisionary scheme as follows:

m o r e of the 10 complete modules indicate the j th mode with probability

2 . 5, S-RETIC i s said to converge on the j th mode. This output convergence

c r i te r ion i s most reasonable i f one a s sumes that S-RETIC always predicates

i t s modal computations on the present sys tem mode, k.

probabilities out of M. become transit ion probabili t ies f rom mode k.

This reduces the equivocation in our output modal decisionary scheme

considerably. As to its neurological versimili tude, motoneurons and

internuncials on which R F outputs of the more modal type play probably

have l ike decisionary charac te r for go, no-go situations.

only a gue s s.

i f 6 o r

Then the

J

But this i s

We now return to the details of formula ( l ) , which ref lects our

M. design s t ra tegies under the two headings:

command, and (b) decoupling. Regarding (a), the f function s e r v e s to

exaggerate the probabilistic modal indications of vector components which

pass through them to a degree determined by the extent to which these

indications depart f r o m the neutral . 25 point.

rapid overall computational convergence by amplifying the differences

between the 1st ' 2nd, 3rd, and 4th modal component gains around

(a) redundancy of potential 1

Thus the f functions promote

26

I I I I I

I I I I

~

I I I I

f(P)

2 . 8

. .

. 1

f(p) = 2 . 8 1 I I I I I I I I

= 6 . 2 5 ~ - 1 . 575 I I I I I I I I I I I I I I

I I I I I I I I

I I

I I I I I I I I I

I i I I

I I I I I I I I .7 P I! 0

L 1 I

1 1

a I >

f(p) = -. 4 Figure 8, The f(p) function.

27

interconnected M.

differences between the corresponding p , p6 modal probability values.

One consequence of this, as we sha l l see , is that s o m e Mi tend to pick up,

o r recru i t , other m o r e equivocally indicating M. over to t h e i r modal

persuasion by a logic s t r ikingly para l le l t o the frequency domain logic

Mi2? . . . , M. , M. loops in accordance with the l1 'N '1

i a. 1

j

J

3 ' ' mediated by manifolds of coupled nonlinear osci l la tors . This is redundancy

of potential command.

Also in connection with s t ra tegy, a), we identify the most c ruc ia l

M. as those whose output p. vectors have components with values fur thes t

f r o m . 25. We ca l l such vectors "peaked. ' I The C Ca, and C fac tors 1 1

6 rr2 and p

' P a i 6 i i Tr of each M. a r e always 1, 1, and 1 if the corresponding p

1

a r e not peaked (do not have component values differing great ly f r o m . 25).

But if at any time instant ( i . e . , computation clock t ime) , any of the f(pT ), i

f ( p ), o r f (p ) components are 2 1 o r I O , C

2, o r 2 respec'tively for that time instant (The a symmet ry between C

C , and C was necessa ry because of the two other f ac to r s in

, C , o r C6 are set to 1. 5, =2 a S. i a

Tr 2, a 6

C = C C Q). Thus, for example, if any M. h e p and p vectors i 1 TT. a =1 =2 1

Tr

near ly equal to (. 25, . 25, . 25, , 25) and a p6 -7 (. 7, . 1, . 1, . l ) , p, and

This Pa. 1 6 i again is redundancy of potential command.

S-RETIC decoupling following overa l l {vi} s ignal configuration changes.

The purpose of this s t r a t egy is t o prevent S-RETIC f r o m being e i ther too

trigger-happy for new modal computations after sl ight and unimportant y

changes, o r too prone t o lock forever on output modal indications that ge t

deeply intrenched at S-RETIC's output. (Monkeys and pigs have the m o s t

tr igger-happy and sluggish R F s we know of among the higher ve r t eb ra t e s . )

The idea ;s to quench a and 6 s ignals after s ignif icant y. changes t o

sufficient degrees and for sufficient durations t o allow injections of new

yi,-derived M. output s ignals into the a and 6 streams.

in CT. If S-RETIC is converged on mode j at t - 1 and t h e r e a re any {Vi}

i i cannot overwhelm p ' s proper effect on M. ' s output p. vector.

1 1

The decoupllng s t ra tegy, b) involves both a loca l Mi and global

1

1

The b) s t ra tegyls global decoupling is expres sed by the Q fac tor

I I I I, I 1

I I I I I I 1 1

I

28

changes f r o m t - 1 to t, Q is increased by an amount and for a duration

that is roughly proportional to the degree of entrenchment of S-RETIC in

mode j a t t - 1.

table:

The values of Q a t t a r e determined f r o m the following

number of Mi for which the jth component of pi 1 . 6 5 at t - 1

value of Q a t t Q a t t

0 to 3

4 t o 5

6 to 8

9 to 10

3 . 0

3 . 5

4 . 0

4 . 5

Q is decreased by 1 each t ime s tep af ter t until it reaches a minimum of

1, and then it remains a t that value until the next overal l {y.} change.

a t any time, on the basis of a new {vi} change, a new Q is computed which

exceeds the value Q has decayed to f rom the las t Q computation, then, and

only then, is Q se t to its newly computed value.

the (1 ) formula of M3, M4, . . . , and M1 2.

If 1

The same Q is used in

The b) s t ra tegy 's local decoupling is expressed by C

determined separately for each M. according to the following table:

. It is ,l

1

Number of p t i changes CT1 value

f rom t-1 to t that a r e > 1 at t

C

is 2 instead of 1 in o rde r to keep M . ' s output p. normally about equally

dependent on pli and (p'Ii, p'lli). rule

occur s at t = 0, when an S-RETIC simulation run begins.

At t = 1, the next t ime step,

is handled just like Q each t ime s t ep af ter t, except that its minimum =1

1 1

An exception to the foregoing C, 1

Then CTF = a. 1

is r e se t t o 1.

29

Our concept of the s t ra tegic difference between local and global

decoupling r e s t s on the following analogy: Suppose a board of 1 2 medical

doctors, each a general is t as well as a specialist in some different a r e a

of medicine, has to decide on which of 4 possible t rea tments each of a

s e r i e s of patients should receive.

S-RETIC module: each patient's medical record corresponds to one

S-RETIC overall {vi> input; and each t reatment corresponds to one

S-RETIC output mode. Every t ime the board looks a t a new patient, the

1 2 doctors a l l decouple the i r decisionary t ies to havea separa te look a t

the records. A f t e r that, they begin their discussion, and in our terminol-

ogy, information couple back in with each other in an attempt to shake sown a consensual decision (legal sense of consensual),

to global decoupllng in S-RETIC.

Each doctor corresponds to one

This compares

Now suppose that the whole board, save one, in passing f rom the

ith to the ( i t 1)st patient i s unable to see any significant differences in the

records , The one that is able to should be left alone long enough to a r r i v e

a t some preliminary conclusions of his own, and then he should be given

a special opportunity to gainsay the r e s t of the board 's tentative conclusions

until they have heard him out.

process of board discussion and play his regular role in shaping a total

board decision.

module.

decoupling, is that it promotes grea te r overal l decis ionary speed because

of i ts bet ter organization and g rea t e r efficiency.

overall decisions a r e not in general a s soundly derived, because the i r

startpoints a r e not neutral.

After that, he should submit to the f u l l

This corresponds to local decoupling in one S-RETIC

The value of it, when contrasted to j u s t additional global

The pr ice of it i s that

We now turn to the details of the h, T, and h - l blocks in F igure 7.

Their effect is to r e s to re

significance of p ' s components is not great ly dis tor ted in the process .

Since differences between smal l 6 probability components (e. g., between

. 2 5 and . 0 5 ) a re general ly more significant then equal differences between

large 6 probability components (e. g., between . 65 and . 85), we f i r s t pas s

p componentwise through an exponential h(x) of the f o r m shown in F igure

9. W e then add the absolute magnitude of the mos t negative result ing component to each resul t to get a l l componen t s r 0.

to a probability vector such that the relative -

-

We finally pas s the

30

resu l t s of this componentwise through a " t ranslated inverses9 of h(x),

denoted h-'(x), a s shown in Figure 9.

a r e derived a s follows:

The equations for h(x) and h - l ( x )

The refore

ax ' a l . 6 - 1 = 1. 6 = e yIg::, (x') = e - 1.

log e 2. 6 a -: - 1 7

Therefore

Therefore

~ ' ~ ( x ' ) = 1. 6 - y I (1. 6 - x') g +

i a(1. 6 - X I ) I yIh(x') z [ I . .- 4 - e - 5

Therefore r--- 1.4a -ax .- ';I e

Note that in addition bo h, T, h - l , N's normalizing and mode-

interacting effects, h(x) limits the influence any given M. can have.

one pathological M. cannot bully the r e s t of the net.

Thus, 1

1

31

X

X '

' t /

Figure 9. h(x) and h-l (x) curves.

32

Our last M. s t ra tegy is necessary only for speeding up convergences L

in cer ta in c l ea r cut cases , and for enabling convergences when there is no

preferent ia l p' vector provocation towards any one mode. In the la t ter case,

a modal decision mus t be made in default of any determining input.

s t ra tegy i s mechanized by adding in at the 15th nonconvergent cycle of each

modal computation that gets that far without converging, the multiplication

of each jth component f curve by

i The

1 2

i = 3 12 C

i = 3

4 I= (jth component of p.) 1

j = 1 , 2, 3, 4. G. = J

(C of a l l components of p.) 1

This could be analogous to the development of diffuse decis ionary field

s t rengths in r ea l R F s .

Before describing our S-RETIC simulation resul ts , we shal l t r y

to impar t some intuition on what to look f o r in the following pre'cis.

RF.: modeling problem is fundamentally one of appropriately matching:

(1) the se t of all possible correlated overal l RF?: model inputs; ( 2 ) the

manner in which the RF::' model ' s regional (i. e . , Mi type) logic allows

initial ascending and descending ( i . e . , a and 6 type) signal sequences to

evolve through i t during a modal computation; and ( 3 ) the nature of the

possible sequences of changes out of the correspondents to A, C, and the

S. in F igure 5, and a l so the nature of their associated sequences of modal

spec if icat ions.

Our

J

Beyond this, it is important t o emphasize a few basic organizational

and operational aspec ts any sat isfactory S-RETIC model, denoted Retic

below, m u s t have. F i r s t , it mus t have sufficient input scope with respect

t o the overal l cen t ra l nervous sys tem (CNS) model in which it res ides s o

that i t can receive the c ruc ia l S. information in every eventuality; and it

m u s t have sufficient computational capacity s o that it can a r r i v e a t the

right modal decision, regard less of whether o r not conflicting o r

competitive demands appear over different A, C, and S. sys tems. F o r

it is established that r ea l re t icular formations must be able to cope with

vir tual ly eve ry possible sufficiently cor re la ted ba r rage of input signals.

L

J

3 3

Second, a Retic must keep its flow of computation close enough to its

input receiving a r e a s s o that a l l input changes can quickly exer t the i r

influence over its output and modal calculations; and the m o r e important the

input changes, the m o r e quickly and profoundly must these influences be

exerted.

which momentary delays and wrong outpds can mean fai lure o r annihilation.

Thus, unlike some cort ical systems, Ret ics must be pre-eminently

interruptible, and not given to long periods of indecision because of

excessive logical depth.

modal commitments too quickly,

to noise and meaningless distractions (like "dreams, for example). Third,

the logical design of a Retlc m u s t be extremely economical.

heavy decisionary demands placed upon it would make it too large and too

slow. Aside f rom a Retic 's conditioning, habituation, long-term learning

plasticity, and spatio-temporal coding of information, it i s essent ia l ly a

combinational logic circuit with very many highly cor re la ted inputs and a

sma l l number of possible stable outputs. The main economy of any Retic

organization of the general type suggested in Figure 6 s t ems f r o m its

repeated use of a fixed amount of modular logic throughout each modal

computation. That is, logic signals a r e recirculated f r o m combinations

of M. units to combinations of M. units a t success ive time instants during

each modal computation until an actual o r approximate decisionary

equilibrium is reached.

the modal outputs a r e produced. In general , such a scheme enables the

logic of each Mi to be used at near ly full capacity throughout each modal

computation, and a l so enables each Retic input channel t o be monitored

continuously. This is vastly different f r o m the way conventional one-

way-flow combinati.ona1 logi; nets work in engineering sys tems.

This i s only reasonable In command and control sys tems fo r

Yet Retics m u s t not be allowed to compute new

for this would make them too vulnerable

Otherwise the

1 1

Then the computation is said to be complete and

To recapitulate, a Re t l c must be a wide, shallow, anastomotic

logic net, consisting of a logil-ai he te rarchy of r a the r tightly coupled and

s imi la r computlrig modules, each the equivalent of about one neuron deep.

34

5 . SIMULATION RESULTS

Appendix 9 contains selections f rom our S-RETIC simulation data.

The "Run number i, cycle number J, sigma se t number k" byline appearing

at the top of each page r e fe r s respectively to which of th ree selected

simulation runs the accompanying data were obtained from,:$ which Zi of

the run is currently under tes t , and which computational pass through the

{ M1 } since the introduction of the present E. the data per ta in to.

i, j, k t r iple will henceforth be denoted [ i, k, j ] . the 5-tuple of y values entering the M. in question. The "normalized

p-pr imes" a r e the present p' vectors passing f r o m the first to the second

pa r t of each Mi.

the p. vectors out of each Mi. On the f i r s t -cyc le page of each Z. in the pl

columns we have underlined the most significant high values and encircled

the mos t significant low ones. Column sums for all p ' s and p.s a r e given as

indications of column averages, and though never underlined o r encircled,

they a r e always important in the determination of modal decisions.

r e f e r to the 10-tuple of p. vectors on the f i rs t -cycle page of each E. as the

initial conditions for that Ei .

This

"Template" r e fe r s to 1

1

The "modal probabilities at end of present cycle" a r e

1 1

1

We

1 1

Since much of the meat of this report i s contained in Appendix 9, the r eade r is encouraged to look at it for himself--i t won't take him long

i f he follows only the most significant high and low values.

design s t ra tegies all contribute to S-RETIC circui t actions a s intended,

so we shall not so r t them out for separate discussion. Rather we shall

summar ize the most important features of S-RETIC behavior.

The various

They a re :

(1) S-RETIC always converges, and always ( so far) in l e s s than

2 5 cycles. Every t ime modal gain factors were used, S-RETIC'S overall

modal preferences were clear ly established in their relative degrees over

the first 14 cycles of the Ei. See [ 1, 3 , 141, [ 2 , 7 , 141, and [ 3 , 1, 141.

The purpose of the gain factors is to speed up, and in r a r e cases enable,

convergences f r o m 14-cycle start points of approximate but inexact standoffs

between 2 o r more highest modes.

:$ In this paper we include the first run only. in the corresponding Michigan State University Division of Engineering R e s e a r c h Report.

All th ree runs a r e included

3 5

(2) Once S-RETIC converges anew af ter a t l eas t one previous non- convergent cycle for a C

maintained a previous convergence, though, for a few cycles a f te r a C i change, then deconverged, and finally reconverged to the same o r a

different mode a few cycles later. See [ 3, 4, 1-41. Because S-RETIC

always converges and then s tays converged in this sense, we could redefine

an S-RETIC mode to be that stable region of operation entered into a t the

final convergence of the overall input in question.

it s t a y s converged. S-RETIC has a t t imes i'

( 3 ) After convergence, S-RETIC's pi vectors always head fo r a

limiting se t which never contains any vector consisting of th ree 0 components

and one 1 component. An exception to this would occur i f all p' vectors

had probability 1 in the same mode; but noise in the module T-circui ts

would cure even that. See run 1.

(4) S-RETIC rol ls over f rom one mode to another easi ly and quickly under strong provocation. As this provocation becomes weaker and

weaker, Z. initial conditions, and gratuitous S-RETIC circui t par t icular i -

ties, play a larger and l a rge r role in the corresponding modal decisions.

"Strong p' provocation" can mean high component values for one mode and

low ones for the others ; o r it can mean a high?' column total on one mode

and low ones on the others ; o r it can mean a blend of both.

component values appear severa l t imes in each of s eve ra l different modes,

we have a dissociated (or disintegrated) situation, and the corresponding

modal decision is often determined by the C. initial conditions - and sometimes

in a surpr is ing way [ 1 , 5, 1-93. The logical complexity of a modal

computation (as judged by us ) is usually about proportional t o the number

of cycles before convergence unless p' provocation is weak o r equivocal

( see [ 2, 7, 1-20]). By ' ' logical complexity'' we mean the degree of p'

competition and conflict, and the intricacy of the pat tern of modal unbalances

among the pa . and pgi vec tors a t the s t a r t of a modal computation.

1

2

If high:'

1

A

-.

1

(5) Appendix 9 contains severa l specific decis ionary effects that will now be discussed:

36

I

In [ 1, 2, 1071, we see a nice resolution of competition, o r

conflict, between module p' vectors. Here th ree mode 4

.5-components overcome a 1 and a .4 mode 3 component.

This is as it should be; an M. should hardly ever , i f ever,

be absolutely cer ta in of an overal l response mode jus t on the

bas i s of i ts own y input.

1

In [ 2, 1, 1-51 we see M recruit ing other M. over to its

persuasion against the p' averages. 5 1

-L

In [ 1, 5, 1-16] we see th ree mode 1 . 4 s overcome a higher

mode 4 '$I average and initial condition bias ( see column

totals for the G . in [ 1, 3 , 13). 1

In [ 1, 5, 1-93 we see a higher mode 1 initial condition bias

overcome a higher mode 3$' average and an impress ive

list of mode 3 $ ' peak values.

dissociation, o r general scat ter , of high component

values here , we would be displeased by the convergence

to made 1. A s it is, it seems pleasingly bio-logical. Note

that it i s difficult to discern any r ea l computational p rogres s

in cycles 2 through 5, other than a slight reduction of

var iances among the mode j components of the p

j = 1, 2, 3, 4. This prompts us to challenge anyone

to wr i te a se t of decisionary motion equations for S-RETIC.

The outcome for this Xi a l so convinced us that S-RETIC

could still su rp r i se u s a f te r hundreds of hours of studying

its behavior.

If it were not for the

2

i'

This is one justification for simulating

S-RETIC.

-L

In [ 3 , 1, 1-1 91 w e see the p' mode averages swamping out two . 7 mode 1 p' peaks. F r o m (c), (d) and a case similar

to the one just considered but not included in Appendix 9, we conclude that either $' peaks, o r p' averages, o r pi

initial conditions can c a r r y an S-RETIC modal decision in

oDDosition to the other two aligned against it.

A

It can a l so

3 7

happen that any two of these can c a r r y a decision in oppo-

si t ion to the third, See [ 1, 61, [ 2, 3-51, and [ 2, 7-81.

( f ) In [ 3, 1, 1-1 91 we see a contest belxeen exactly matched

modes 2, 3, and 4 decided by gratuitous circui t par t icular i -

t i es (noise of a kind). In this c a s e S-RETIC was little m o r e

thar, an elaborate 3-state flip-flop, s tar ted at a neutral point.

(g) In [ 2, 1, 1-10] we see a boundary M. 's (M21s to be prec ise)

mode 4 pl peak c a r r y a modal decision against strong mode

1 initial conditions, a competing mode 1 peak, and a second

highest mode 1 'pl component average.

1 A

( 6 ) We remark that anyone who carefu l lyperuses a significant block

of our simulation da ta will note many other interesting facets of S-RETIC

decisionary behavior.

modal component among the M. determines to within a tolerance the

regenerative gain of that component each cycle; the degree of "dissociation"

(or the prevalence of apparently uncooperative phenomena, amenable to,

say, a simple statist ical description) among the 3. is not par t icular ly re la ted

to the variances of their jth components; and Mi "recruitment" is - not jus t the

converse of p. inhibition by other 8..

F o r example: the ra te of aggregate swelling of a

I

1 -

A

1 1

( 7 ) Our simulations have proved that S-RETIC pa rame te r settings This i s fortunate, for real R F s function all the can have wide tolerances.

way f rom coma to convulsion.

Appendix 10 gives a macro-level flow char t for our simulation

program. It w a s writ ten in a v e r y t ransparent MAC language b y J. Blum,

and run on the MIT Instrumentation Laboratory Honeywell 1800 Computer.

38

6. FORM-FUNCTION RELATIONS FOR S-RETIC

I

S-RETIC computes an output modal function of the twelve p1

and a set of initial condition vectors , IF. The vectors pll, p12,. . ., p

IF vectors a r e completely specified by the pll, pI2, p13, . . . pIlz vectors

at the end of the previous Zi.

F(pI1, . . . , p'125 I ), and its possible values 1, 2, 3, and 4.

I 12'

W e denote S-RETIC'S output modal function

F F has three symmetr ies that we shall re ly on throughout the

remainder of this Section:

I.

11.

111.

F o r large numbers of modules, where the smal l number

combinatorics of the M. -to-M. connections no longer 1 J

appertain, F is invariant under a l l p' and corresponding

I vector permutations, i # 1, 2. That is, simple exchanges

of M. positions within the net, each M.'s y connections

remaining intact, do not affect F.

i

1 1

F implies the s a m e evaluation function on a l l four modes.

F sometimes converges on a mode different f r o m the one

with highest average over the p1 vectors, o r the IF vectors, i o r both the p t i and IF vectors. See 5-(e) of the previous

Section. More generally, our simulation data show that

S-RETIC can converge to a mode favored by ei ther the p' i average, o r the IF average, o r the pli peaks (as in . 7 , . 1,

. 1, . 1, e. g. 1, in opposition to the other two of these factors

aligned against it. Any two of these factors aligned together

can a l so overcome the third in opposition; and any one of

them can overcome the other t w o when they a r e against it

but not aligned on the same mode against it.

a strength-of-effect symmetry over these three factors.

There is thus

In the r e s t of this Section, we outline an argument to show that

S-RETIC computes a mode function, F , that no S-RETIC net without a

and 6 connections but with nonlinear summative output scheme could

compute even though it be allowed more equipment. We denote such

a n al ternat ive net, E, and picture one in Figure 10. The zi in Figure

39

1

- 0

0

0

y lines

8

0

Z 1 2 3 4

Figure 10. A modular net without intermodular coupling.

40

10 correspond exactly to the y-to-;'. par t of each S-RETIC module, except

that we allow each M. to have more y inputs and g rea t e r logical complexity

than its M. counterpart.

that at least two conflicting

(y} inputs.

the Section. ) Obviously the larger S-RETIC is, the eas i e r this condition is

to satisfy.

1

1

We insist, though, that the Gi be simple enough so 1

tendencies arise for each of s eve ra l overall i (This condition cannot be made m o r e prec ise until the end of

We next suppose that Z computes a 4-valued modal function, Z, of

its present and next-to;last $ I

the function computed by Z, Z(pI1, . . . , ptk, Iz). than o r equal to the number of corresponding S-RETIC modules.

is defined to be 1 , 2, 3, o r 4 according as

input sets. W e denote the latter set Iz, and i W e require that k be less

Z , then,

1 s1 = c f i j (PIij) i, j

s2 = f 2 i j (PIij) i, j

= c f3ij (p ' . . ) 'J i, j

s3

1 5 i 5 k; i an M. module number 1

(2)

1 5 j 5 4 ; j a mode number

k has the highest value, where p'

a r e a r b i t r a r y continuous functions of their arguments (proportional t o log

functions, perhaps ) .

is the jth component of i, and the f i j i j

If we had let our p ' . .s in Figure 10 be general degrees-of-presence - 'J

of various propert ies in N 's overall input stimulus, instead of specifying

them as we did, F igure 10 could be reduced to a nonlearning Pandemonium

Machine o r t o one of s eve ra l popular Bayesian logic designs in the special

case.

obtaining a rigorous comparison between fl and S-RETIC.

such a comparison, though, would cer ta inly have been s imi la r to the one

we a re undertaking if it could have been made.

We did not allow this because it would have prevented us f rom

The resu l t s of

41

Let us now derive our underlined statement above. It is not a

theorem because pa r t of our argument for it stems f r o m observations and

extrapolations on our simulation data. Our method will be t o t r y to equate

F and Z in the special ca se where IF is null (recall that fo r the first overall

input of a simulation, C = a), the boundary $'i of S-RETIC both equal . 25,

. 25, . 25, . 25, k = 10, and the Z set of $ I i is identical to the F set of $ I i in

eve ry case.

why, and then generalize t o get our result .

fk . . = fk employ the same evaluation function on each mode.

a cyclic permutation, II, of the $ I i components such that

T

W e will see that Z cannot equal F under such conditions and

Since Z m u s t be invariant under all nonboundary $ I i permutations,

in equations ( 2 ) for all (i, I ) pai rs , 1 5 i, I 5 10. z m u s t a l s o 'J I j

Thus the re m u s t exist

m the same f o r a l l 1 5 i 5 10; and such that fp i j (pIij) = f ik (pIik) f o r all

all unnecessary indices, equations (2) become

i and fo r I # m if and only if II ( P I . . ) = p' and ll (pl iP) = p 1 im. Dropping 'J ik

and

s1 = c f . (PI . . )

s2 = f j (II(PIij))

s3 =

:: c f . ( ? (P I . . ) ) s4 J 'J

J LJ

2 J '3

c f . (II ( P I . . ) ) ( 3 )

42

Our S-RETIC simulation data requires that in each S s u m of (3), k f be monotonic increasing and f j # k, be monotonic decreasing, , k j ' fk

though, cannot increase too fas t a s a function of p'

average p'. could not determine Z ' s value as often as it does F's.

the f . cannot decrease too fas t a s a function of p

In other words, symmetry I11 gives us a s eve re se t of constraints on the

f . s . We denote the bounds on f and f . for S by Sup f and Inf f . (see

Figure 11). These bounds must be established f r o m simulation data.

because if it did the ik' A Also

1

for the same reason. J ij

J k J k k J

Suppose now that we have a se t of 3'. vectors a l l equal to (. 25 t E, 1

. 25, . 25, . 25 - E), a corresponding la rges t S in equations (3 ) equal to

S1, and a corresponding S-RETIC F value equal to 1.

just large enough to make this t rue (in our S-RETIC simulation this was

about . 03). Next, consider a second$' vector set comprising

PI1 - - . . . = pI8 = (. 1, . 3, . 3, . 3 ) and pIg = p t l 0 = (. 7, . 1 , . 1 , . 1).

equations (3), the difference between S1 for the f i r s t and second$'i s e t s

is

k We want E to be

i In

A S = 1 8 [ f l (. 1) - f l (. 25 t E ) ] t 2 [ f l (. 7) - f l (. 25 4- E ) ]

t 8 [ f 2 (. 3) - f 2 (. 25)] t 2 [ f 2 (. 1 ) - f 2 (. 25)]

t 8 [ f3 (. 3) - f 3 (. 25)] t 2 [ f 3 (. 1 ) - f3 (. 25)]

t 8 [ f 4 ( ' 3 ) - f4 ( . 2 5 - E ) ] t 2 [ f 4 ( . I ) - f4 ( - 2 5 - E ) ]

1 1 - - a 1 t a 2 -

1 1 - a 3 t a4

1 1 - a 5 t a 6

1 1 - a 7 t a8 Y

43

fk

t 0 -

*.i J

m P;j P;j

Il Figure 11. Bound curves on the fk and f . functions for Sk

in equations (3 ) . J I

1

44 I

where a l l a! 2 0. Similarly, 1

k k A s k = - a 1 t a 2

k k - a 3 + a 4 ,

k k - a 5 t a 6

k k - a 7 t a 8

k = 2, 3, 4

In our S-RETIC simulation, the F value for both se t s of $ I . vectors 1

above was 4.

Z to bes t approximate F - - indeed, perhaps equal it. Certainly Z # F if

a l l A S = 0. Also, since the average of the second?' s e t above is (. 22,

. 26, . 26, . 2 6 ) , the f . could not be such a s to make each S equal t o the

average over i of the pl

has many $' .-set solutions for any selection of f . functions, the question

a r i s e s a s to whether some second:' vector set other than the one given i above exists such that e i ther all AS fo r it a r e 0, o r the highest Sk for i t k does not correspond to its F value. If so, Z # F.

Suppose we have a se t of f f o r equations ( 3 ) that enables j

k i

J k Fur thermore , since each 4-tuple of S values k ik'

1 J

We a r e cer ta in there does not exists a s e t of f . functions for J

equations ( 3 ) such that Z could be made equal to F. In fact, given any

alleged se t of such f , we could at least a lmost always find a second

$ I i set such that a l l A S k = 0 for that set but F 's value changed between

that set and the f i r s t $ I i set (under the assumed null conditions on I

and Iz, of course). Anyone who, studies our Appendix 9 should have no

difficulty in seeing this. But if that should fail, we could always find a

second$Ii s e t such that F # Z for it by concentrating on ak adjustments

in (41, which because of the bounds and signs on the f. J

directional, smooth, and simple. This should be evident.

j

F

j would be uni-

It now follows ra ther easily that removing our argument r e -

quirements that the number of Ei be 10 and that the Z cti se t equal the

F ; I i s e t would not change the character of our resu l t a t all. Nor can

45

I

we see how it could be refuted using S-RETICs with m o r e and m o r e modules.

They would s e e m only to demonstrate it even more spectacularly.

the underlined asser t ion ear ly in this Section.

Hence

46

-

I

1 I I 1 I

7. CONCLUSIONS AND FUTURE WORK

W e can safely infer f rom our simulation resu l t s , the cooperative

effect of our design s t ra tegies , and S-RETIC's specifications (Appendix

8), that proportionately increasing the numbers of everything but modes

i n S-RETIC would improve i t s performance in every important respect .

Given a l a rge number of Mi, we would put those with most similar area

central i far thest apa r t in the model.

a r e a central i to form, at the (yi} level, the equivalent of a highly over -

lapping cover of the complete {v i} bundle.

of much g rea t e r decisionary acuity and competence, and vastly g rea t e r

invulnerability to M. failure, than the present t e s t model enjoys. To a good approximation, such a s t ructure would be slower than our test

model according a s the ratio:

We would a l s o want the s e t of Mi

This would give u s an S-RETIC

1

Number of splittings of each output l ine - N1

N2 - -

Number of Mi

i s lower than 3/12, its value in our simulation.

More than about 6 modes might slow the model down considerably, o r make it unduly sensit ive to noise.

This a s sumes 4 modes.

How does the complexity of S-RETIC inc rease with l a r g e r { u i }

bundle s i z e s ? First, l e t us assume an S-RETIC with k modules, 1

Y i S jRETIC, k l / m = ( 1 2 x 5)/9 < 7.

factor i ly remain l e s s than 7m

If so, N /N

all S-RETIC complexity proportional to m.

with the corresponding exponential relation in switching theory.

inputs to each module, and a {u i) set of m lines. In our simulated

We believe that k l / m might satis-

(constant) for increasing k, 1 , and m.

above would increase l inearly with m, giving u s an over - 1 2 This compares favorably

We have in the foregoing supplied one paradigm for getting a family o r more than two information-coupled automata to work together

in a slightly biological fashion.

by biology and is a good command and control computer for some purposes,

we make a c la im for bionics.

To the extent that our resu l t was inspired

47

We especially emphasize that S-RETIC i s not jus t a glorified

pat tern recognition net.

of a real- t ime R F model.

peculiari t ies and strengths as compared to a l a rge c l a s s of nonlinear,

modular, Bayesian logic nets.

It satisfies the additional temporal constraints

The previous Section indicates its functional

S-RETIC's compound virtue as a computer is that it is fast ,

economical, reliable according to the redundancy of potential command,

and operational on all of its inputs at each t ime step.

We have recently augmented S-RETIC as shown in F igure 12

The w j

to begin our study of possible R F time-binding mechanisms.

l ines there ca r ry crude a r e a central is information f rom the i r modules

of origin.

r emember its two previous inputs, outputs, Rpc reinforcements, Rac

reinforcements, and modes of overall convergence. These combina-

wi tions a r e then used to modify the p' vector response to future y i , inputs.

tioning, extinction, habituation, and long- te rm adaptation among the M..

The problem is to get a group of M

appreciates the overall input-overall output correspondence problem, to

l e a r n in an integrated, harmonious fashion. The main obstacle s e e m s

to be interference due to local signal ambiguity on overall input-

overall output relations. A major by-product of this work so far has been

that we can now see how to engage and drop out R F operational pa rame te r s

in a gradual manner.

on this.

There a r e 1 3 mi and 7 y i per Mi. We allow each Mi to j

j This enables u s to real ize several types of cooperative condi-

1 each of which only partially i'

We a r e indebted to W . Brody for severa l insights

We would like to note the Scheibels' suggestion that S-RETIC

might be a more valid model i f we regarded our M. as instantaneous - functional instead of fixed-regional R F subcomputers.

this r e m a r k seriously, fo r it implicates the legit imacy of our s impli-

fications, the appropriateness of our outlook on R F s , and the propriety

of our linguistic level.

Finally, we inquire as to the actual value of our simulation.

1 We a r e taking

Its

main justification is that we can now think with ou r S-RETIC model, and

not jus t -- about it.

partially derive some new insights into R F c i rcu i t actions.

We hope his will help us to partially invent and Another

48

justification is that w e now know the prec ise consequences of interlacing

our simple se t of s t ra tegies in a behavioral mechanism, and we can see that these consequences w e r e much too complicated to apprehend before-

hand.

mathematics, as symbol manipulation by logical rules , is good for steady

s ta te and microcosmic brain processes , but not yet global decisionary

ones. Nothing e l se could

possibly serve, w e think, where each little cause can have such major

effects, yet where each person is still able to make so much sense of

it.

The simulation has strengthened our prejudice that c lass ical

We need a type of scientific poetry for that.

This takes us back to K. Craik's , "The Nature of Explanation.."

He would say, with K. Popper and others, and we would agree, that we

do not yet have a theory of the RF. invalidate our claims; our concept has not yet produced any r isky predic-

tions; it does not forbid any measurable R F event; and we have not yet

proposed any real alternatives.

up, no out.

F o r there is no experiment that could

In this sense, ou r resul ts were though

We wish to acknowledge Dr. Michael Arbib as a major consultant

on the organization of this report.

might have to discussions with Drs. A. and M. Scheibel, Dr. L. P roc to r ,

and Dr. W. McCrumb. D. Peterson and E. Craighill ass i s ted greatly

i n programming the simulation, and R. Warren was a ready contributor

of program organizational ideas.

W e also owe much of what c lar i ty it

The computer simulation was done at the Massachusetts Institute

of Technology Instrument ation Labor ato r i e s under National Aeronautic s

and Space Administration Contract NSR-22-009- 138.

work was done under Air Force Office of Scientific Research Contract

The theoretical

A F -AFOSR - 10 23 - 66.

49

Converged \ b e , R R Retic Response

Rpc, Rac Reinforcements toS-RETIC y boforo and oftor corrvorgonco tospoc tivo ly.

I i 1 1 1 I I 1 I

I

1 I

I

Figure 12. Augmented first part of typical S-RETIC Mi.

50

I

I I

1 I

I I I I I i I I

I I D I

8.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

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5 7

APPENDIX 1

Convergence on a Model

To a cer ta in extent, any chunk of nerve t i s sue that has to per form

both an analyzer and an integrator function, a s R F * does, can be viewed

a s an assembly of coupled nonlinear oscil lators. (In a ve ry r ich sense,

a l l neural t issue amounts t o such an assembly, and cer ta inly behaves non-

l inearly overall. ) In fact, the variety of abductive logic our R F * employs

on its highly correlated input sequences strongly suggests a model of

ra ther tightly-coupled multi-stable oscillator units cooperating probabilis - t ically s o a s to admit a t any given time only one of a smal l number of

possible stable overall operating modes. Thus we turned to Weiner 's work

on correlation-coupled nonlinear oscillators, which shows that there a r e

forbidden zones about each stable point, and suggests that generally such

sys tems behave a s required. 126 To illustrate, F igure A l - 1 (Wiener's

F igure 8.4 in reference 126)depicts for such a sys tem the shape of the

probability distribution of oscil lator frequencies about a normalized stable

point, as calculated by Wiener.

far a s our R F * model is concerned is that there is enough variability

about the stable point t o permi t flexible sys tem operation.

we found that a central defect of all such sys tems is that there is no

reasonable analytical o r experimental way of determining anything basic

about the t ransient behavior between stable mode points following input

changes.

a r t i f ic ia l neural net theory we know of to data.

r e f e rences 11, 12, 18, 35, 107. ) But to be able to follow such t ransients

is cen t ra l to our task, s o we had t o regard these sys tems as useless.

Recalling that single nonlinear oscil lators a la I ~ i i n o r s k y ~ ~ behave too

rigidly, and that l inear sys tems a r e unable to exhibit the necessary

memory , cell assembly, and modal features, * we abandoned a l l coupled-

osci l la tor and neural-net approaches to RF?\ theory construction a s

u t te r ly hope les s.

The crucial thing about Figure Al-1 a s

Unfortunately,

The same holds t rue for every sufficiently complex nonlinear

(See, for example,

% In other words, the long-term responses of l inear sys tems a r e de te r - mined in a 1:l manner f r o m their input drives, whereas this is not t rue for nonlinear systems. So in this sense, l inear sys tems a r e irredundant, whereas nonlinear ones have a chance of being redundant in the right way f o r modeling neural behavior.

A1 -1

l-4 I

a a, u 5 k a,

W a, k

fi

* a0 a, k 3 M

Fr

.r(

.

.d

v1

k a,

- 5 3 .d

. 4 I d

4 a, k 5 M

Fr .d

Evidently we required considerably m o r e initial s t ruc ture than they

would afford, i. e . , we needed a well-developed se t of s t ra tegies for de-

signing a skeletal model of R F * behavior.

computer- s imulation and mathematics to investigate the complex behavioral

consequences of varying these strategies.

without doing too much violence to the biology, we returned to the Scheibels'

stack of poker chips analogy.

Then we could r eve r t to a

In o rde r to pursue this plan

The resul t is described in Section 4.

I I I I I I I I I

I

Al-3

APPENDIX 2

yk Function Table

z ( F . A I J . ) V ( U . A ~ ) v ( m A IJ.) 'k L J J I I 1

k 1 j I k i j I

1

2

3

4

5

6

7

8

9 10

11

12

13

14

15

16

17

18

19 20

21

1

1

1

1

1

1

1

1

1

1

1

1

2

2

2

2

2

2

2

2

2

2

2

3

3

4

4

5

5

6

6 7

7

3

3

4

4

5

5

6 6 7

8

8

8

9 8

9 8

9 8

9 8

9 8

9 8

9 8

9 8

9 8

22

23

24

25

26

27

28

29 30

31

32

33

34

35

36

37

38

39 40

41

42

2

3

3

3

3

3

3

3

3

4

4

4

4

4

4

5

5

5

5

6 6

7

4

4

5

5 6 6 7

7

5

5

6 6

7

7

6 6

7

7

7

7

9 8

9 8

9 8

9 9 9

8

9 8

9 8

9 8

9 8

9 8

9

A2-1

APPENDIX 3

Some ai; F, C, yi relationships for E, with only seven ui and the

y. comprising the se t of all 35 3-variable symmetr ic switching

functions of the f o r m y. = (u. A uk)v ( u k A ul)V (ur A uj) . 1

1 J

Number of the seven Number of ul, . . . , u7

this can happen

Number of the 35 u i which equal 1 combinations for which yi which equ'al 1 in

each of these combinations

7 21

35

35

21

7

A3-1

0 5

13

22 30

35

i

3

4

5

6

APPENDIX 4

- t o Mi Connection Table 'j

j

1 14

23

32

37

13

24

31

38

41

4

11

15

20

40

6 17

28

29 33

i

7

8

9

10

5 8

21

25

34

11

10

16 26 27

39

12

12

18

23

37

4 2

9 13

18

28

36

7

10

1 9 30

35

3

15

22

32

39

1

12

26

34

41

3

6 7

19 30

A4-1

APPENDIX 5

Exemplary {vi} , Table (6 Mi) th

A p' vector corresponding t o overall ordered

5-tuple of y. set of yi. input signals 'j inputs to M6 J

'6 '6 2y6 3y64y6 p'4

00000

10000

01 000

11 000

00100

10100

01 100

11100

0001 0

1001 0

01 01 0

11010

001 10

10110

0110

11110

00001

10001

01 001

11 001

001 01

101 01

01101

11101

0. 273

0.625

0.563

0.250

0.250

0.357

0.833

0.500

0.000

0.500

0.692

0.583

0.125

0.333

0.444

0.400

0.000

0.000

0.333

0.250

0.083

0.200

0.500

0.640

0.152

0.125

0.375

0. 250

0.350

0. 214

0.167

0.167

0.308

0.000

0.154

0.167

0.333

0.000

0.222

0.200

0.040

0.333

0.222

0.375

0.083

0.250

0.167

0.120

0.333

0.000

0.000

0.250

0.150

0.071

0.000

0.167

0.385

0.333

0.077

0.083

0.292

0.167

0.111

0.200

0.440

0.111

0.333

0.167

0.417

0.250

0.333

0.120

0. 242

0.250

0.063

0.250

0. 250

0.357

0.000

0.167

0.308

0.167

0.077

0.167

0. 250

0.500

0.222

0.200

0.520

0.556

0.111

0.208

0.41 7

0.300

0.000

0.120

A5-1

Appendix 5 (continued)

0001 1

1001 1

0101 1

11011

001 11

10111

01111

11111

0.000

0.000

0.214

0.200

0.333

0.000

0.000

0.212

0.000

0.000

0.143

0.400

0.000

0.000

0.750

0.182

0.750

0.500

0.357

0.200

0.333

0.438

0.125

0.364

0. 250

0.500

0.286

0.200

0.333

0.536

0.125

0.242

A5-2

1.

ci A -

0- 1' u2, * - * , u9

0 o . . . o all 9 tuples

APPENDIX 6

THE PREPARATION SCHEME FOR APPENDIX 5

y 5 -tuple into

M1 2 . . . . M1 M2

00000 00000 00000

First of all we constructed a char t of the form:

0 0 . . . 0

all 9-tuples

-

Mode assignment

point of zi

1 0 0 . . . . 0

using the y-function char t of Appendix 1, and (actually, several

different) assignments of Xi points that we found

"interesting and reasonable" se t s of p vectors for enough Z

to enable us to per form a meaningful simulation.

1 yielded enough 2

lT i

2. Then we constructed a chart of the form:

1. After much labor . This aspect of our simulation design is currently one of the most difficult and crucial.

2. Cf. the text Section on Simulation Results.

A6-1

APPENDIX 7

Mi - to - M. Connection Table J

1 Connections into

the jth module, for j =

8

9 10 11

12

I

th Let a.

module and go into the jth module.

It ca r r i e s the k component of p which we denote

. Below we list for each a. and 6. the i of the Pik Jk Jk corresponding p . This gives the module of origin

ik of the connection.

o r 6. in the figure above come from the i Jk Jk

th i'

a a a a 6 6 6 6 j l j 2 j 3 j 4 j l j 2 j , j4

9 2 4 11 11 4 10 1

8 10 10 6 3 12 8 5

6 7 7 10 7 1 6 9 2 9 3 12 5 8 9 4

11 6 5 9 8 5 12 10

3 5 2 7 10 3 5 12

5 11 6 4 6 10 11 3 3 4 7 1 8 12 3 9

10 4 12 2 1 6 3 7

7 8 8 8 9 11 7 11

A7-1

APPENDIX 8

Distribution of the li - j I in the Mi - t o - M. Connection Table J

li - j l

1 2

3 4 5 6 7

8

9 10

Number of

M. -to-M. 1 J

connections with this

1 i- j l

18

16

12 1 1

9 7

6 5

3

2

Ideal distribution to satisfy pr (an M.-to-M. 1 Co J

connection with l i - j I = k) = pk - Gli - j l

- A s

where Co is a connection constant such that 10 Z

k= 1 88 pk = 88, with roundoff to the nea res t

integers.

18 15

1 2

10 8 7 6 5 4 3

88 = total = (8 x 10 f r o m Mg through M

( 4 f r o m M 1 ) ’ t ( 4 f r o m Mz) ) t 12

A8-1

APPENDIX 9.

SIMULATION RESULTS

A9-1

3 9

4 3 a v 4

Q 0 9 ?l

In m (0

N

L n N .s N

co r- m

N

3 z

.t 0 + m

3

0

N

3

n 0

,s-l

0

4 0

9l

0

4 1 0

m

0

4 4

0 - I t m 0

r ) 9.

0

d

0

m

0

0

N

0

n. 0 4 m 0

0

0

IU

0

n. 0 ut 4

m d

0

J '3 3 5_

Y

3 4

9

m 3

4 0 -4 m

0

0

N

0

n 3

m 0

-4 3

m

0

-4 0 n N

0 d

0 m 0

0 3 .-I

0 4 m

a 0

3 3

a 0

d

0

N n 0

:v .-+

m

m

3 0 a

0 z 0

0 0

0 0

t- rl ,n

3. F 3 n c.l

.3

21 'n

0

0 4 * 0

n c 3 .-l

0

c) -4 n

0

n 4 .cI

0

n 4 n

0

n 4

0

c)

21 n

0

0 4 m

0

0

0 .n U

0

0 :\1 -. m

t 4

3

0

3

. 3

N

3

n ? 3

a 3

3

0, 3

0 a 0

% 2 3

3

3 3 c

3

d

3

3

- 3

N n

3

3

3

- 3 n N

0

3

9

N 0: L d

I 3 Z

n 3 3

.AI

U u t - u

-1

I

w t 0 0 0 0 3 0 3 0 3 3 3 0 . J) J -I a a F r

c c ..I

II 'I

l-i I-

n w w

n m

a 9 . 4

3

N . 4

* . n 9 & n L n . n 9 L n o 1 n 3 9 d m 4 4 4 m c r L n c r l n 9 m o (n m ,m i, m N m N a . . . . . . . . . . 0 0 3 0 0 3 0 0 0 0 N

r n ~ & n t n i n ~ n o m r > 9 d m . 4 d 4 m d r n 4 & 9 r n o m m m o m ~ m ~ .t

0 3 3 0 0 0 0 0 0 0 N . . . . . . . . . . .

.AI U z LLJ 0 w > 2 3 'd

a m o m m m o m o m ~ o n i n ~ m t n o 9 o o o r o o n J 0 3 3 3 0 0 0 0 . . . . . . . . m in 0

0 . 0 .n

N . 0

0 0 Q 0 N

I- L ?

3 4

m 0

0

0

(U

0

n 0

0 0 0 0 0 3 3 0 A D 4 - 4 4 a e . n m - t m m m d m ~

0 0 0 0 0 0 0 0 . . . . . . . . 0

4

0 m

0

0 N .-4 . rn

3 0 d n m N 0 . 0 0

0

N

0

n . . 0 z 4 r d L n

J 5 U Ln -

0 d n

0

n .n w . 0

0 N *

0 m

3

N n . 0

0 N

w ! - 0 3 0 3 0 0 0 0 ~ 3 3 0 < J) J a F

-1 4 c

d

c Y

I- In w r y c w

in I: 2 z

9 9

0

N

0 d

0

d C O Q N O I O a D F Q m O

O O O O O O O O O N

0 0 0 0 0 0 0 0 0 1 3

m m u m m m m a m ~ . . . . . . . . . . P- ry 9

0 0 d

,AI n 3 r

Q

-4 0 a

0 d

n ~ r - m d m ~ m m m m In 9 d r C . * * t 9 0

0 0 0 0 0 0 0 0 0 0

N : 2 o d O C ( d N d . . . . . . . . . . nJ n J 9 0

4 c z W 0 X w > ? 0 U

m O l n m d 9 m c c ~ N m ~ r - r c a ~ o a m . n m a l n t n a m e m 9 F m u m

0 0 0 0 0 0 0 0 0 0 9 . . . . . . . . . .

O m 0 0 0 ~ F ~ l n 9 N ~ fi m a n 0 * - .s N m J N N O Q O m O d n J . . . . . . . . . . . . 0 0 0 0 0 0 0 0 0 0 0 0

* N d

0 m

d

3t Z d

n

In 4

N . m

0 9 0 m 3 N 0 Q F d D O ~ ~ G D ~ C P O J ~ ~ N O N & n N O N 9 0 9 ~ U ~ 0 . . . . . . . . . . . . 0 0 0 0 0 0 0 0 0 0 0 0

.s OD 9

m 4 J 0 0 z

d c 4 -I a 5'

c r- m IL1 CI I11

.. t

c rrlr * ... 2 z z 3 U

9 9

0

N

d

d

a W

9 J N U

OD * I n I c- w

m r m u

U

3 z

a u u s

2 z n

c

0

N

0

n

0

N a 0

n

0 In

0

v

3

N a

0

n

m a 0 ’ 2 1 0 0

N m a 0 0

0 3

0

m 0

0 n N

a 0

0 0 3 0 4 N

a m 0 0

-4 3 ln a

m

I) m * m

3 9

9 9

0

N

d

d U

r- 9

N

m

(0 co N

N

" r t-

I

o, 9

N

4

0 z

N

cy

z 2 a

Q 9

0

N

0 d

0

d

0 W

Q J N U

m v (0 J L n

I I-

W

m >

U

~

Y W 0 AI > z 0 J

a

J

D I U 2 0 w o

P c

3

d 0 0 r

rY 0 LI

c m .

W I n d

0 n N

0 0

0

N

0

n 0

0 m N 0

0

0 n N

0 0

0 u 3 0 4

d

0 3 0

0 0

0 3 0

0 d

0 0 n

0 0

3 0 0 e 0

d

0 0 4

m

1c

0

0

- 1

0

* In d

0 0

a0 rn .n

0 0

4 m 1-

0 0

0 0 0 4

J

0 3 m

0 0

0 m N

0 0

0 In Y

0

0

3 0 *u e 3

0 u r )

4

Ln

0 ut m 0

0

3 J U 0

0

0 .r 0

0 0

0 0 0

3 0

d

9

m 3. N

0 0

m N 4

0 0

N OI N

0 0

In h m e

0

0 3 0 d 4

r-

c) 9

0

4 0

0 9 0

0 0

0 n .t

0

0 n N e

0

0

0 0

3

4

w

D >*

0

m 0

0 In N

0 0

0 m N

0

0

0 0 N

0 e

r )

CJ d

0 r4

c) 3 In

0 0

0 0

0

.-l 0

c) In c)

0

0

0 an m 0

e

0 u 0 rl 4

d 4

0

m 0

n. 0

0 3 N

0 0

0 0 m

0 0

0 n 4

0 0

0 C ) 4

N d

e m m 0

0 0 .t

N 0

m m d

0

N

.t P 0

N 0

9 9 a

0 a N

d

4 .t .t

m .o m

O U J

w o z z n

a LJ A m U

m * w (c or 4 a

N

w 9 - 1 N U m > m u OD . t m

w v a u o c r

9 0 or

-. I e

a 4 w

U z LLl u w > z 0 U

a 4 4 0 o w

0 I

E D

4

9 m 4

0 z

U 0

N a 0

n 9

m

0

- I

0

0 3 3

yc

0

0

c

0

0 a m 0

0

0 N In

0 0

m 3.

m 4 0

0 0

0 01

a 0

4

0 3 1

0

m 0

0 3. In

a 0

N a 0

a 3

N 0 n N

0 a 0

.r an

a 0

0

N n 0

.r 4

In N

Q)

N n

0 9 0

0

N n 0

0 0 9 0 n N 4 d

a 0

d a 0

a 0 z -

0

0 0 0 0

a 0

a 0

0

0 a a 0 m

c J ln n

in U

0

0

m m *n

0 0

0 J\ ry

0 0

n 4 7

0 0

N m N

0 0

n 0 9

0 0

n 9 .+

a 0

c) In Y

a 0

0 L n n

n 0 * 6

r v m .-.

0

0 0

0

3 0 0

a 0

0 n N

4 n 0 a

0

3 0 3

0 0

-4 -I N

a 3

0 0 N

VI L

m 0

0

$ 3

a 0

rl

3

N n

0 0 N

3

m n

0 4 n -t 4 m 0 N

0 Y ;J

0

3 a

0 a

0 0 0 0

sl J V

U P v u

c . 4

c . ._

d .4

r) 9

9 9 . d 0

N .

d

W .o J N U m * m u Q urn

L

W 0 IJ > z 0 U

a

0 z

4

.3 LLI

w o 5

z n

c

s 0

N

0

n a

0

N

0

n e

c) JI Y . 0

3

?1

3

n .

m m 0

- 0 . 4 n m . 0

m I) 4 . 0

4 ‘0 c3

3 .

N Q F Q O N O ’ 0 9 9 ~ O * 0 3 r O I n ~ - N 4 m m m ~ ~ ~ a t . t ~

0 0 0 0 0 0 0 0 0 0 . . . . . . . . . .

4 O h n U ’ d U ’ J N d m ~ ~ n ~ r n s m j r c r n u \ 4 * N O d O * 4 O O . . . . . . . . . . 0 0 0 0 0 0 0 0 0 0

m 0 @ . m

N P In

N 0

d

In P . F4

m P- a U’ In N m In N 0 N u m d m m N r c n J s 0 , 0 O O N O d N d r- 0 0 0 0 0 0 0 0 0 0

rn A

. . e . 4 ? ? . 8

0 3. 0

0 . 0 3 0

e d

0 0 n

0

3 3 ‘3

0

r-

(3

0

- 0 . 3. In 4

0

m m ‘n

0 .

+ * ,w 3 .

0 3,

0

m . 0 ln N

e 0

n In U

a 0

,2 3 ry

0

0 v m 0 . 0 3. 4

0 . c, s

e 0

3 0 0

3 .

P 3 N

0 . In U

d

4 . N 0, 21 . 0

In

m 0

- .

(D * I m 0 . a N

e 0

n

0 0 0

e 0

4 r 0 - 4 . 0

0 9

0

4 . 0 P 0

0

0 3D .? . 0

3 n N . 0

0 0 m

0

0

N

0

n . 0 m W

0

3 3 N . 3

0 3

0

an . 0 0

0

4 . 0 hn n 0

0

m 0

n .

0

m 0

n. . 0 3 N

0 . 0 0 rn

0 e

7 %n d

0

4

r m 0 .

Q .n u m

0

m m .n . N

4 4 m N .

.4J t

w

t 9

9 9

0

N

. d . 4

U

.

m c o ~ ~ o c r m o m m ~ f- ( ~ a ~ u m e c r u m u N o m - a o r . o r n - o o In . . . . . . . . . . . 0 0 0 0 0 0 0 0 0 0 d

- I t

w Y c I;J U Y W > z 0 U

. . . . . . . . . . . 0 0 0 0 0 0 0 0 0 0 r4

4 P rc\ . m

N o ~ o * o o m m Q o o o O . n O n I n b N O D h n 3 3 N m O d N * d N O N d N . . . . . . . . . . . . 0 0 ~ 0 0 0 0 0 0 0 0 0

m n

m . . s

c

0 - 0 a I O 0 N 0 0 0 0 0 m ~ o m m U m O o ~ l n 0 'u 4 n n r\i n Y o .+ ~1 n m . . . . . . . . . . . . 0 0 0 0 0 0 0 0 0 0 0 0

.m

.n . N

m

.O 3 ~ 3 - + 0 0 l n U 3 0 3 0 n n o r n o o h 3 a o n n N O O N N O ~ - I N N ~ ~

0 0 0 0 0 0 0 0 0 0 0 0 . . . . . . . . . . . . d 4

9 - 3

N . a

4 il I 3 z

J

c

D t

n 0

9 9

0

N

. d . d

.?

II

J

0 r n

9 N m m (0 U

c d J V F U

c Z W 3 [L W

0 U

s X I- d

d Q 0 5 cz

0

w c . J

F

c

n

w

c) n N

0 . 0 ;n N

0 . c) m N . 0

0 n N

0 .

0 1 2 0 n d

d

. . . . . . . . . . . 0 0 0 0 0 0 0 0 0 0

O e L N e e N N N + O 9 . ? t J F a D d N L n P N . ? a0 S ~ ~ ( ~ ~ ( N ( U ( U I U ~ Ln . . . . . . . . . . . 0 0 0 0 0 0 0 0 0 0 N

c ~ o m r - m a ~ r n o m d OI ~ ~ t ~ m n m m r - e m m 9 O ~ ~ O ~ O ~ O O O m

0 0 0 0 0 0 0 0 0 0 + . . . . . . . . . . . d 9 S r - r c r D L n m * a 0 n . t . t + m ~ ~ ( l r o ( l r m + 0 0 0 0 N 0 0 ~ ~ . l l 0

0 0 0 0 0 0 0 0 0 0 4 . . . . . . . . . . 0

0 33 0

0 . 0 0 0 . d

0 0 0 . 0

0 0 0

0 0

d L.1

d 0

0

h b

0

0 .

U n d . 0

9

? . 0

d .m N

0 .

0 J 0 .-l

.?

0 3, 0

0 . 0 In N . 0

c) In N . 0

0 0 N . 0

0 3 d 4

In

0 V I m 0 . 0 .T * 0 .

0

n

0 . 3 3 0

0 .

d

9

9 3 1

N

0 .

In N d . 0

N CI, N . 0

Ln > m

0 .

0

0 U

4 d

r-

(0 $ 1 m 0 .

rn n N

0 . 0 0 n

0 .

.?

4 . 0

s rl

P

0 01

0

d . 0 9 0

0 . 0 33 .t

0 . 0 .n N . 0

0 C ) 0 0 -4

OI

0 3 m

0 . 0

N

0

n . 0 rl\ Y . 0

3 0 N

0 .

rl c, -4

0 r(

0 3 In

0 . 0 0

0

r( . n

n In . 0

0 n m

0 .

0

0 3

4 4

P I 4

0

m 0

n. . 0 0 N

0 . 0 0 * . 0

0 n

0

4 .

0 3 4

N d

d > m . m

9 m .j

m .

rn

.I,

N

m . 4 d m . N

J) -I .I c

c

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APPENDIX 10

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