Kybemetik 10, 58-60 (1972)@ by Springer-Verlag 1972

Holographic Aspects of Temporal Memory and Optomotor ResponsesA. BORSELLINOand T. POGGIO*

Istituto di Scienze Fisiche, Genova, ItalyLaboratorio di Cibemetica e Biofisica del CNR, Camogli, Italy

Received August 2, 1971

Abstract. A mathematical analogy between the holographicmodels of temporal memory and Reichardt's optomotor theoryis stressed. It is pointed out that the sequence of operationswhich is essential to any holographic model of brain function.ing is actually carried out by a nervous structure in the opto-motor behaviour.

Some implications in both the optomotor theory and thehypothesis of neural holographic processes are further sug-gested.

The optomotor response of insects (their evokedresponse to movement relative to themselves in theirvisual surroundings) allows, through a quantitativesystem analysis, the specification of some fundamentalfunctional principles of the central nervous system(Reichardt, 1969).

It is not necessary for our purpose to specify allthe experimental and theoretical details, otherwisewell known (see for example Reichardt, 1957). Highlyreproducible experiments lead to undoubtable, con-clusions about the existence of physiological mecha-nisms able to connect sensory input and motor outputin a well defined manner.

A model, devised by Reichardt, accounts for thefunctional properties of the neurophysiological system.The model is a network of linear illters, the weightingfunctions of which are fitted by analyzing the experi-mental data. It allows a very accurate prediction ofinsect's responses to previously untested stimuluspatterns.

The model (Fig. 1) envisages two cross.connectedinformation channels from two input units A and B(the elementary event which evokes an optomotorresponse consists of a sequence of two light stimuliimpinging on two adjacent receptors). The two inputelements transform the light stimulus into the timefunctions LA and LB which are determined by thestructure and speed of the pattern. The functions L.Aand LB are further transformed by a succession ofcomponents that carry out linear transformations,multiplication and time averaging.

If we designate with WDFand WDH the weightingfunctions on the channels and with WGGthe auto-correlation function of the light-flux fluctuation G(t),then we obtain-according to the model and on thebasis of all the experimental conclusions-the followingexpression for the optomotor response

00 00

R(L1t) = J WDP(n) dn JWDH(~) WGG(n-~ -L1t) d~

0 0 (1)00 00

- J WDH('Y})dn J WDP(~) WGG('Y}-~ -,1 t) d~0 0

the equivalent of which in the frequency domain is00

R(At)= ! f YDYl (YpYIl- YJ Yu) 8(w) eiwdldO). (2)-00

* Present address: Max-Planck-Institut fUr BiologischeKybernetik, Tiibingen, Germany

YD, Yp, Yu are transfer functions of the illters and8(w) is the spectral density of G(t), related to WGGthrough the Wiener's theorem.

It does not matter if G(t) is a periodic stimulus ora random distribution of light points:. in both cases thepredictions fit perfectly the experimental findings(Reichardt, 1969).

The exact localization of the anatomical structuresunderlying the optomotor behaviour is still an un-solved problem. Many aspects however are now quiteclear.

Some histological sections of the first optic gan-glion -the lamina -have sometimes been considered asan indication that the pattern of fibers from retinato lamina may be isomorphic to the information flowdiagram of Fig. 1. As a matter of fact the projectionof fibers from neighboring ommatidia in one elementof the lamina (cartridge) is well compatible with theReichardt's scheme. Braitenberg (1967) and Kirsch-feld (1967) demonstrated that this localization cannotbe upheld any more. While it is true that fibers fromdifferent ommatidia come together in one cartridge,

Moving intensitypattern

A

YLA

$

B

YLa

E;JLinear filters

Ommatidia

L*A L*a

Linear filters

)lH' ~H~

~ ~L'AF' Lt!H L"BF.L\H

$ $

Multipliers

Low - pass filters

L'AF' L'SH L'BF' L'AH

L'AF . L~H - L:'!3F. L1H Subtraction units

+R

Fig. 1. Reichardt's model describing stimulus-reaction rela-tions of optomotor responses(from Reichardt, 1969)

10. Ed., Helt 1, 1972 A. Borsellino and T. Poggio: Holographic Aspects of Temporal Memory 59

these fibers collect information from one point of theenvironment only, detected by six retinula cells in sixdifferent ommatidia. Consequently the projectionbetween retina and lamina is such that an individualcartridge does not receive information from two dis-tinct points of the environment, a necessary conditionfor any correlation model (Reichardt, 1969).

The lamina however can not yet be excluded as apossible localization for movement detection sincea precise pattern of lateral interconnections betweentriplets of cartridges have been recently discovered(Strausfeld and Braitenberg, 1970). Although thetopology of these connections is in principle the samethat would be required by the minimal G6tz's (1968)motion detector model, some preliminary findings byKirschfeld (1971) suggest other connections too.Kirschfeld had just succeeded in obtaining optomotorreactions by stimulating selected pairs of rhabdomereswithin one ommatidium. Actual combinations,. effec-tive in eliciting optomotor responses, seem to requireother connections in addition to those found byBraitenberg, possibly in the lobula. Only furtherstructural work at this level will answer the problem.

We will now consider the Reichardt's model in itsmathematical aspect only. From this point of viewEq. (1), characterising optomotor behaviour, showsthe same mathematical structure underlying the mostgeneral holographic transformation, namely convolu-tion of some function with the autocorrelogram of akey signal. We particularly notice that the holographicmodel of temporal recall proposed by Gabor (1968a,1968b) is mathematically identical to Eq. (1). As animplementation of the holophone concept-first intro-duced by Longuet-Higgins (1968)-Gabor suggestedand discussed (Gabor, 1969) a general analogue in thetemporal domain of the association properties ofholography. In particular Gabor presented two two-steps transformations imitating the recording-recon-struction sequence in holography. The first of them isa succession of a convolution and a correlation: theway of making a record is to convolve the key sequenceA with B. To recall B, we form the cross-correlation ofthe record with A', where A' is a fragment of A.

Essentially the Gabor's model requires the recordof

~B,d~) = JB(T) A(~ -T) dT

that is, with the convolution symbolism,

~BA =B *A

where B and A are temporal sequences and ~A istheir convolution product. In the "recall" we form thecross-correlation of the sequence A' with ~AB:

F(t) = JA'(~) ~BA(~+t) d~.

Thus the Gabor's "recall function" becomes

F(t) = J B(t -'YJ) d'YJJA(x + 'YJ)A'(x) d x

that is, in the symbolic formalism;

F=A'@(B *A) =B * (A'@A).

If A' @A ~ ~ (A is a noise-like function) weobtain F(t) ~ B(t) (recall). If A' @A =J=b (A is not anoise-like function or A' is not a fragment of A), weobtain a filtering operation formally identical to theoptical holographic compensation and synthesis pro-

-

cesses that have been developed in the spatial domain(Stroke, 1969).

A very interesting task, Gabor suggested, is tolook for evidence of nervous structures capable toperform this scheme of operations. Processes such asthose Gabor described may be actually present in thenervous system, but no evidence supported the hypo-thesis of their existence.

Now, we are perhaps able to acknowledge thevalidity of Gabor's suggestion. As a matter of factwe can rewrite Eq. (1), disregarding the physical model,as

R(LI t) = Jd~ Jdy [JVz>F(y+~) JVz>H(~)

- JVz>H(y+~)WDF(~)]q)aa(y-Llt)

where y ='YJ-~.Defining

W(y) = J [~F(Y +~) JVz)H(~)

- ~H(Y+~) WDF(~)]d~

(8)

(9)

we obtain

R(LIt) = JW(y) tPaa(LIt -y) dy = W * (G@G) (10)

because tPaa is an even function. If we write Eq. (6)for A =A' as

F(t)= JB(tl) PAA(t-t1) dtl =B * (A@A) (11)

(3)

the formal identity between Reichardt's and Gabor'smodels turns out clearly.

This identity enhances, through behavioural ex-periments, the hypothesis of nervous holography-likeprocesses (Van Beerden, 1970; Westlake, 1970). Atleast it shows that the sequence of operations requiredby the holographic or holophonic models of temporalmemory (Longuet-Higgins, 1968; Gabor, 1968a, b;Watson, 1971), is actually carried out by a nervoussystem in the optomotor responses. This statementseems thus to suggest the existence of nervous struc-tures with functional holographic properties and givesstrength to Greguss' hypothesis that biolographicprinciple is not restricted to the echolocating activity(Greguss, 1968).

In this respect it is appropriate to notice thatReichardt's model is in biology the only one able todescribe the functioning of a complex nervous struc-ture with great accuracy and in a formalized language.That is why the holographic analogy appears to be aquite interesting one. Fig. 2 is then a formal compari-son between optomotor model and a holophonic mem-ory device. Though there are possible differentschemes of holographic memory, let us carry out theanalogy outlined in Fig. 2.

The holophonic scheme of Fig. 2 stores first the" key" f with the signal W; afterwards it will recallthe time function W, when excited by a fragment of I,if the sequence is noiselike. On the other hand thelight-flux fluctuation, which in the optomotor theorycharacterizes the insect's panorama, can be regardedas the key signal. Moreover it turns out that the phys-iological environment of the insect, by means of itsmovement, generates a light-flux function which isproperly noise-like. Therefore the movement percep-tion of the natural surroundings recalls the filter func-tion W which may well represent some genetic built-inmemory. Thus this formal comparison enables us to

(4)

(5)

(6)

(7)

-----

60 A. Borsellino and T. Poggio: Holographic Aspects of Temporal Memory K ybernetik

a) b)

f'

B* (A@A')W* (f@f')

Fig. 2. Holophonic (a) and optomotor (b) formal schemes

think at the optomotor network as a kind of short-term holophon-like memory. .

Anyway it is to be emphasized that the meaningof the holographic aspect of the optomotor responsesdoesn't lay on the property of distributedness, perhapsthe most important for the hypothesis of holographicmechanism within the brain (Westlake, 1970). Forthis reason it seems to us that the suggestion of neuralholographic processes (Longuet-Higgins, Van Reerden,Westlake) appears overall meaningful because of itsmathematical structure more than because of its

physical mechanism. For instance the same mathe-matical holographic framework is able to underlie theoptomotor behaviour as well as some general processesof filtering, compensation and associative recall(Stroke, 1969).

Therefore the same principles may operate at theperipheric and the central levels of nervous system.According to this point of view the formal identitybetween Gabor's and Reichardt's models essentiallysuggests that the characteristic sequence of filtering andcorrelation may be fundamental to neural processes.

All the "physical" holographic implications-dis-tributed coding and wavelike mechanisms (Van Reer-den, Westlake )-are not confirmed, and they wouldrequire of course more direct evidences.

In conclusion we have shown so far that the same

general formalism underlies both the optomotor theory

and the holophonic model of temporal memory,suggesting deeper neural analogies. As a consequencethe experimental investigation of optical data pro-cessing in the insect's nervous system will perhapsprovide also a ground for the analysis of patternrecognition and memory.

Acknowledgments. One of us (T.P.) wants to thank theEuropean Molecular Biology Organization (EMBO) for ashort term fellowship allowing him to visit the Max-Planck-Institut fiir Biologische Kybernetik, Tiibingen. Helpful dis-cussions he had in this occasion with Prof. Reichardt and hiscoworkers are gratefully acknowledged.

References

Braitenberg, V.: Patterns of projection in the visual system ofthe fly. 1. Retina-Lamina projections. Exp. Brain Res. a,271 (1967). .

Gabor, D.: Holographic model of temporal recall. Nature(Lond.) 217, 584 (1968a).

- Improved holographic model of temporal recall. Nature(Lond.) 217, 1288 (1968b).

- Associative holographic memories. IBM J. Res. Devel. la,2 (1969).

GOtz,K. G.: Flight control in Drosophila by visual perceptionof motion. Kybernetik 4, 6, 199 (1968).

Greguss, P.: Biolography, a new model of information process-ing. Nature (Lond.) 219, 482 (1968).

Heerden, P.J., van: Modelsfor the brain. Nature (Lond.) 220,177 (1970).

Kirschfeld, K.: Die Projection der optischen Umwelt aufdasRaster der Rhabdomere im Komplexauge von Mnsca. Exp.Brain Res. a, 248 (1967).

- Personal communication (May 1971).Longuet-Higgins, H.C.: Holographic model of temporal recall.

Nature (Lond.) 217, 104 (1968).Reichardt, W.: Autokorrelationsauswertung als Funktions-

prinzip des Zentralnervensystems. Z. Naturforsch. 12b, 448(1957).

- Movement perception in insects. Processing of optical databy organisms and machines. Proceedings of the internationalSchool of Physics E. Fermi, CourseXLIII (ed. W. Rei-chardt). New York: Acad. Press 1969.

Strausfeld, N. J., Braitenberg, V.: The compound eye of thefly (Musca domestica): Connections-between the cartridgesof the Lamina Ganglionaris. Z. vergl. Physiol. 70, 95 (1970).

Stroke, G.W.: Principles of holography. New York: Wiley1969.

Watson, C.J.H.: Plasma echo and spin echo holophones.Nature (Lond.) 229, 28(1971).

Westlake, P.R.: The possibilities of neural holographic pro-cesseswithin the brain. Kybernetik, 7, 129 (1970).

Prof. A. BorsellinoUniversita ill GenovaIstituto ill FisicaViale Benedetto XV, 5Genova/Italy