.:;" Physics Essays volume 12, number 4, 1999

Complementarity Principle and Cognition ProcessSisir Roy and Menas Kafatos

AbstractIt is generally believed that the cerebellum's function is to help the brain coordinate movements,but the recent neurophysiological evidence challenges this dogma. Apart from being considered aspecialized control box, the cerebellum participates in many activities of the brain, includingcognition. Here we examine the correspondence between response and percept domains. It isproposed that there exists a principle similar to the complementarity principle in quantummechanics that operates for response and percept domains.

Key words: complementarity, response domain, percept domain, quantummechanics, quantum filters, cognition

1. INTRODUCTIONIt is generally believed that the cerebellum's function is to

help the brain to coordinate movement. (I) Many studies onboth animals and humans have supported this idea. But therecent neurophysiological evidence challenges this dogma. (2)

The evidence indicates that apart from being just a special-ized control box, the cerebellum participates in manyactivities of the brain including cognition. It is of interest toinvestigate the correspondence between the response andpercept domains. It is proposed here that there exists aprinciple similar to the well-known complementarityprinciple in quantum mechanics that operates for responseand percept domains.

The multistable state of perception as well as the varioussolution properties of response lead us to introduce the ideaof quantum filters operating in these domains. Quantumfilters are generally used for selective measurements inquantum mechanics. For example, we may consider aStern-Gerlach arrangement where we let only one of thespin components pass out of the apparatus while we com-pletely block the other component. More generally, weimagine a measurement process with a device that selectsonly one of the eigenstates of the observable A and rejects allothers. This is what is meant here by selective measurementin quantum mechanics. It is also called filtration because onlyone of the eigenstates filters through the process. Filters arecommonly used in the domain of optics. Interferometers anddiffraction gratings are the instruments to disperse light intoits constituent wavelengths. Filters used in interferometryare devices to transmit wide bands of wavelengths. Varioustypes of filtering are investigated in modern optics.PGeneralizing the concept of filter fromthe domain of optics,we can think of a filter for selective measurement as itapplies in the context of Stern-Gerlach experiments inquantum mechanics. Although filtering measurements areconceivable classically, the main difference between the

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classical filter used in optics and the quantum filter used forselective measurements can be envisaged by studying whatpertains to incompatible observables. This will be elaboratedon in Section 4. In fact, the essence of quantum mechanicsis that there exist incompatible observables in microphysics.This concept helps us to build a unified structure for under-standing the conceptual problems related to cognitionprocesses. It must be emphasized that we are taking the ideaof a quantum filter at the conceptual level in order to betterunderstand the cerebellum function. We are not consideringany quantum process that mayor not be operating in someregions of the cerebellum, at least at the present state of ourunderstanding of the brain function. The motive of this workis to show that concepts such as the principle of comple-mentarity, nonlocality, quantum filter, etc., may play animportant role not only in quantum mechanics but also inother branches of science. It was Niels Bohr himself whoproposed that complementarity extends beyond the atomicrealm. (4) This extension has been proposed as a fundamentalstructuring principle in constructions of reality. (5) In accor-dance with these views, in this paper we concentrate onextending complementarity to neuroscience.

2. SPATIOTEMPORALREPRESENTATIONOFIMAGEIn order to recognize a particular object, the brain must go

through a matching process to determine which ofthe manyobjects it has seen before best matches the object underscrutiny. Moreover, an object may appear in differentportions, in different sizes, and under various orientations onthe retina, giving rise to different neural representations atthe early stages of the signal system.

Research carried out on associate memories gives us someinsight into how to handle the problem of pattern matchingwith the help of neural networks. On the other hand, toproduce object representations that are invariant withrespect to dramatic fluctuations due to changes in position,

SisirRoyand Menas Kafatos

size, orientation, and other distortions that occur on thesensory inputs, is far from being understood. McCullochandPittS(6) made an early attempt to understand invariantrepresentations from the viewpoint of neurobiology. Theyhypothesized that the brain forms many geometric transfor-mations of the same sensory input and then averages overthese using a scanning process (which they believed was therole of the alpha rhythm) to form an invariant representa-tion of an object.

It can be pointed out that visual attention may provide thekey to forming invariant object representations. The mainidea is that the process of attending to an object places itinto a canonical or object-based reference frame that issuitable for template matching. However, Palmer et al.(7)made no attempt to describe the neural mechanism fortransforming an object representation from one referenceframe to another, because theirs was primarily a psychologi-cal model. Recently, Van Essen and Andersonl'" proposed aneurobiological mechanism for routing retinal informationso that an object becomes represented within an object-based reference frame in higher cortical areas. This is knownas a dynamic routine circuit. The principal idea behind thisapproach lies in the fact that it provides a neurobiologicallyplausible mechanism for shifting and resealing the represen-tation of an object from its retinal reference frame into anobject-centered reference frame. In order to map an arbitrarysection of the input onto the output, the neurons in theoutput stage need to have dynamic access to neurons in theinput stage. Here they proposed that the efficiency oftransmission of these pathways is modulated by the activityof central neurons, which have as their primary responsibil-ity to dynamically route information in the cortex. Withinthis framework, they have further developed their ideas andproposed that visual processing at early stages in the cortexis carried out in a relativistic coordinate system, rather thenbeing simply based on absolute retinal coordinates.l'" Afterthey analyze several interesting situations, it is suggestedthat our perception is based on computations in a relativisticcoordinate frame. It was assumed that an image is repre-sented in the spatiotemporal domain and a fixed velocitywas considered so as to get relativistic coordinates.

It should be mentioned that observations carried out oncells in a eat's visual cortex imply the interchangeable role oftime and position'?' in determining the apparent position ofa stimulus object. Moreover, a certain process of integrationmust occur, because one experiences a single moving object,not a succession of separate stationary ones. But if we takethis integration in a mathematical sense, it must be carriedout in space and time matching that of the moving object,and not separately or in the two domains. Space and timeare thus seen to play complementary roles in image process-ing. This is in accordance with the fundamental wholenessof space- time revealed in its complementary aspects as theunity of space (or "nonlocality") and the unity of time asproposed by Kafatos and Nadeau.i" They assigned comple-mentary status to both spatial and temporal nonlocalities in

the sense that taken together as complementary constructsthey describe the entire physical situation under question,although neither can individually disclose the entire situa-tion in any given instance. In the cerebellum geometry, wehave assumed that a type of nonlocality applies. In otherwords, the cerebellum geometry is described by a Hilbertspace structure. It is worth mentioning that Pribramt'?'constructed a geometry of neurodynamics so that the neuralwave functions can be described by a Hilbert space similar tothe wave-function in Hilbert space in ordinary quantummechanics. Healsoemphasized two important aspects whichshould be considered in brain processes, i.e., that both anonlocality similar to the quantum nonlocality and theHilbert space description for the geometric description apply.In this Hilbert space approach, the cortex is assumed topossess a system of eigenvectors forming a complete or-thogonal system in the processing sense. Here we have alsoassumed a Hilbert space description for the cerebellumgeometry where filters are defmed (see below).

However, much controversy arose during the last decaderegarding the space-time representation in the brain,particularly for the cerebellum. Pellionisz and Llinas(ll)discussed the problems of space-time representation for thecerebellum in a series of publications. In fact Braitenberg' 12)

made a pioneering attempt toward our understanding of thespace-time representation in the brain. Pellionisz and Llinascritically analyzed the situation and found difficulty in usingrelativistic coordinates. In our approach, we have specificallyconsidered a Hilbert space description for quantum filters inthe nomelativistic domain. However, Pellionisz and Llinasgave some intuitive arguments as to how geometricalstructure should be assigned to the cerebellum where bothspace and time are considered simultaneously. On the otherhand, we emphasize that Pribram considered a relativisticapproach for the structure of neural geometry.

3. RESPONSE-PERCEPT DOMAIN ANDOBSERVATION PROCESS

In the spirit of Niels Bohr, Wheeler stated that in theworld of quantum physics, no phenomenon is a phenome-non until it is a recorded phenomenon (in this case imageinterpretation ).

In the domain of quantum mechanics, when the experi-mental apparatus interacts with the observed system, itproduces uncontrollable disturbances. In classical physics,one can control these disturbances in a sense by modifyingthe apparatus and repeating the experiments, and it is inprinciple possible to reduce these disturbances to zero. Butthis is not possible when one has an experimental situationinvolving a quantum system. In analogy with a quantumsystem, image generation always involves the observation ofa phenomenon against some background. In other words,the system measurement device relation in quantum me-chanics has a formal correspondence to the object-back-ground relation in image interpretation. Again, the back-ground introduces ambiguity with respect to the object

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similar to the disturbances produced by the apparatus in aquantum measurement process. An interpretation systemcan, at a given lime, in response terms, only handle adichotomy relation. It seems to be necessary for anything tobe in some sense computable and interpretable that it has toaggregate certain properties, which taken together providea stable background against which it can be related. This isstrikingly similar to the preparation of the state vector inquantum mechanics. What we propose is a formal similaritybetween quantum-mechanical measurements and the object-background relation in image representation (see also Roy etal.(B»).

Everything now indicates that the background defines aframe of reference for interpretation. This is equivalent to arestriction of the domain ofpossible responses. Motivated byengineering principles, Paulin' 14) suggested that the cerebel-lum computes like a Kalman filter. Nevertheless, for Kalmanfilters it is necessary to have feedback. On the other hand, inmodeling cerebellum functions, if we also take its role incognition, then there should be no such kind of feedbackcontrol in percept domain as is observed in motor action. Sothe analogy of using Kalman filters is not very suitable indescribing the percept and response domains. It follows thatthe concept of selective measurement with quantum filtersmay be more appropriate for the percept-response domains.

First, let us define the state vector. The state of a system atany time t is represented by a normalized state vector 'Y (t).This may be a state for a classical system such as the solarsystem, pointers, and Brownian particles as well as quantumsystems such as electrons and photons. This state vector mayalso be used to predict or control a process. In that sense, theKalman-Bucy filter (KBF) is a state estimator for dynamicalsystems as elaborated by Paulin, where he proposed thehypothesis that the cerebellum is a neural analogue of theKBF.

A quantum-state filter' 15) is designed to allow the passageof a particular quantum state and no other. Suppose that thestate I ~> from the eigenbasis A I ~> = ~ I ~> is incident on aP -filter designed to allow the state 1/9 from the eigenbasisPIA> = A IA> to pass. This is equivalent to performing aselective measurement in which only a particular result isacceptable and all others are discarded. Sowe Canintroducea similar concept of quantum filters in the response domain.Let us designate them by the term R-filtering.

What in general can be considered to be the differencebetween classes and symbols on the one hand and responseson the other hand? It seems that symbols and classes havea greater degree of invariance at the price of less specifidty.On the other hand, responses have a greater degree ofcontextual specificity and consequently less invariance. Thisgreater degree of contextual specificity is closely related tothe concept of selective measurements and hence one canintroduce the use of quantum filters in the response domain.

Again, the invariance mechanism for, say, the representa-tion of an object as a combination of views and the responsesinvolved, implies furthermore a form of equivalence between

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structures in the feature domain and in the response do-main. We may say that for the domain of an object, we havea "balance" or equivalence between a particular set offeatures and a particular response. They are, again, equiva-lent because their combination forms an invariant: an entitythat is not perceivable in the combined percept-responsedomain interface surrounding this description. An invariantinevitably implies a combination balance between thepercept and the response domain.

4. COMPLEMENTARITY PRINCIPLE AND PERCEPTAND RESPONSE FILTERS

Recent studies on the role of the dentate nucleus in thecerebellum using magnetic resonance imagtng+"? revealedthat the regions of the dentate involved in cognitive process-ing are distinct from those involved in the control of eye andlimb movement. So it may be that a kind of quantumfiltering is operative in those regions in the sense thatselective measurements are operating. The detailed study ofthose regions may reveal this in the future. This kind ofselectivity also applies to other regions of dentate nucleus inthe cerebellum.

Letus designate the two filters as R-filters and P-fIlters forresponse and percept domains respectively. An illustration ofa quantum-state filter designed to allow only the state Irk>to pass is given in Fig. 1.

Suppose that the state Iuj> from the eigenbasis A I~> =~I~>is incident on the !i'-fIlter designed to allow the stateIrk> from the eigenbasis R Irk> = rk I rk> to pass. This situationis sketched in Fig. 1. The probability that the state I rk> isobtained after passage through the filter given an incidentstate I~>is determined from the usual quantum expansioncoefficients to be 1(~lrk>12. In other words, we have theusual expression

(1)

and when making a measurement of the property repre-sented by it quantum theory states that the probability ofobtaining the eigenvalue rk is the square modulus of theoverlap (rk 1 Uk>' After the measurement, the state is projectedonto the corresponding eigenstate of R. A sequence of twoquantum-state filters designed to allow only the state I rk> ofR to pass is illustrated in Fig. 2.

However, we can equally envisage the expansion of I~>inthe eigenbasis of the P-filter as well, so that

laj) = 2:IPmXPmlaj)'m

(2)

The probability of obtaining I rk> given an Incident I~> cannow be written as

Sisir Roy and Menas Kafatos

Ia J ) = Lip m )(pm Ia J )

~laJ) ~I~_I~oh)

R - filter

Figure 1. Illustration of a quantum-state filter designed toallow only the state 1 rk) to pass.

l(ajlrk)12 =I~(ajIPm)(Pmlrk)12

m

= ~ ~ (ajIPm)(Pm,h)·m m'

We can think of I~) as being "made up" of the states IPm)'and the probability that we are ultimately interested in,1(~I rk) 12, can be viewed as resulting from an interference ofall the possible paths of the kind, namely

Letus consider another situation, where a P -filter is actuallyinserted before the R -filter, as shown in Fig. 2. If the P-filteris set to admit the state IPm)' then the probability P ofobtaining 1rk) given an input I~) is now given by

Let us sum over all the possible filter settings to obtain thetotal probability as

which is not equal to (3) due to the existence of cross termsin (3). These terms are known as the interference terms.

In fact we can rewrite (3) as

where the second term contains the interference contribu-tion from all the possible 1 {Pm}) paths. In the two-slitexperiment, the P-filter is equivalent to determining theprecise trajectory of the particle and (6) is a mathematical

expression of the consequent destruction ofthe interference.Here, in the context of the percept and response domains, wecan regard 1 {Pm}) as all possible alternatives that causeinterference, but the interference will be ultimately de-stroyed when passing through the response domain.

For example, when we recognize an object such as a lineor an edge, we get a localized object. This implies that theinterference-like terms should vanish to allow these localizedobjects to be obtained.

Now, if we look at (6), we get the interference term tovanish if the second item goes to zero, i.e.,

(8)

(3 )

This condition will be satisfied if IPm) is a simultaneouseigenstate of both P and A (and/or R). For this to occur,P and A (and/or R ) must be compatible observables. In otherwords, we must have

[A, .f>] = 0 and / or [lU']= o. (9)

(4)

The action of the P -filter can be detected if P represents acomplementary property to both of the properties.

In other words, a given response in a particular systemstate is equivalent or complementary to a particular percept.

The preceding must be true for all interfaces betweenlevelswhere invariants are formed, which generally must befor all levels of a system. It must then be true for the inter-face of the entire system as well as for its environment. Whatthis implies is that the combination of percepts that a systemexperiences, and the responses it performs, constitute anequivalence, viewed from the wider percept responsedomain. This in turn implies that the entire system as wellappears as an invariant to the environment in a widerdomain. usually a system can be observed externally onlyfrom its response, and the effects in this subdomain are notexpected to be invariant, otherwise the system could notaffect its environment.

An interesting question is immediately raised: Whathappens when there is no response motor action even whenthere is activity in the percept domain? In that case thephase differencebetween the state vectors in the percept andthe response domains may not be 90°, the interference termwill not be zero, and we may have partial interference.

It appears from the above analysis that it is possible todescribe the percept and response domains using theconcepts of state vectors in the Hilbert space and the idea ofquantum filters. There will be an interference term betweenthe vectors defined in the percept and response domains.This is analogous to the double-slit experiment in quantummechanics. Thepercept and response domains are said to beequivalent if the interference term vanishes. It is similar todetecting the complementary aspects like those ofwave andparticle in the double-slit experiment.

(5)

(6)

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----------- ----------------- -

Complementarity Principle and Cognition Process

n n

\a/)u u

P - filter R - filter

Figure 2. A sequence of two quantum-state filters designed to allow only the state Irk) to pass.

In the percept domain, we can recognize an object such asa line, edge, ete. This appears to be a kind of localizationphenomenon analogous to the particle aspect. But in theresponse domain, we have a kind of signal, and a process ofnonlocalization begins. This is similar to the wave aspect. Sothe two domains seem to be complementary to each other.Indeed there might be a correspondence between the perceptdomain and the response domain. We can think of relativis-tic transformations such as the Lorentz transformationsbetween the different frames of reference in these twosystems, as also emphasized by Van Essen. We can think ofa rest-frame in the percept domain and the motion oractivity in the response domain. So there may be a kind oftime dilation that could in principle be calculated using theLorentz transformation. This time dilation causes the phasedifference between the state vectors defined on the perceptand response domains. Then, depending on this phasedifference, the interference term for the filters in the perceptdomain and the response domain will be destroyed.

An interesting situation arises when there is no motoraction but something is there in the cognition process, asfound in recent neurobiological expenments.I''"

It seems that there will be no time dilation between theframe of reference for the percept domain and the responsedomain. Ifwe consider the frame of reference attached to thepercept domain, the phase difference between the twovectors in the two domains now will never be t'P/2 or 3t'P/2 soas to get a vanishing interference term. The vectors may bein phase lag of 0° or 180°.

Here we have a situation analogous to the retaining of theinterference term in quantum processes, so no detection ofevents is possible. But if there is a partial interference (forphase angles less than t'P/2), the detection of differentcomplementary aspects is possible, similarly to the detectionof a hazy path with partial interference. Usually it is notpossible to measure simultaneously the two complementaryaspects with arbitrary accuracy. But one can accomplish thiswith certain unsharpness introduced in measuring thecomplementary aspects. Then the observables are known as"unsharp" observables. This kind of unsharpness may arisedue to various physical situations. For example, in the case ofmeasurement of the spin of an electron, we use amagnetic field

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in the Stem-Gerlach type of experiment, but due to variousphysical factors such as the presence of thermal noise, etc., wedo not in actuality measure the exact value of spin as, e.g., +1/2or -1/2. What we observe is a distribution for spins.

It is proposed here that it is necessary to analyze thedetails of neurophysiological experiments so as to gain moreinsight into the physical causes required to reduce theobservables into unsharp states. As a result, there will be anoverlapping of complementary features. The mutual exclu-siveness expressed by the term complementarity refers to thepossibilities of predicting outcomes as well as specific valuedeterminations. Bohr and Pauli discussed both these aspects.In a recent review, Busch and Lahti(l7) analyzed this situa-tion in detail. They considered two types of complemen-tarity: one is referred to as the measurement-theoreticalcomplementarity and the other as the probabilistic comple-mentarity. The former implies non-coexistence. In the caseof sharp observables, the two formulations are equivalent.However, for unsharp observables, the measurement-theoretical notion of complementarity is stronger than theprobabilistic one. This is due to the fact that in the frame-work of probabilistic complementarity, the simultaneousmeasurements of complementary observables like positionand momentum, or path and interference observables, arepossible. Sowe can construct a joint observable for a coexis-tent pair of unsharp path and unsharp interference observ-ables. Bohr's original concept of complementarity as refersto a strict mutual exclusiveness is thus confirmed for pairs ofsharp observables. On the other hand, Einstein's attempt toevade the complementary verdict can be carried out in somesense. But the price has to be paid by introducing a certaindegree of unsharpness in measuring the complementaryaspects.

5. IMPLICATIONSWe have introduced a generalized principle of comple-

mentarity that applies to structures beyond the quantumrealm as in the case of the percept and response domains.One of the present authors+" discussed the role of comple-mentarity in neuroscience in one of his recent monographs.The recent neurophysiological results'I'" suggest that cere-bellar output projects via the thalamus into multiple cortical

Sisir Roy and Menas Kafatos

areas, including the premotor and prefrontal cortex, as wellas the primary motor cortex. In addition, the projections todifferent cortical areas appear to originate from distinctregions of the deep cerebellar nuclei. Some neurobio-logists' 19) have challenged this kind of distinct behavior ofthe cerebellum by studying some patients. However, theyhave acknowledged that cognitive and motor functions areintegral constituents of the mechanism governing animalbehavior. They analyzed substantial amounts of data thatsupport the view that, if the cerebellum is important forexecuting a specific behavior, it also participates in any long-or short-term modifications of the characteristics of thebehavior. It is clear that there are many ambiguities regard-ing the various experimental results. More refmed experi-ments and critical analysis of the data are needed so as toclarify our understanding of the cognition and control ofmovement. The problem arises regarding the integrationprocess so as to coordinate dentate regions that are responsi-ble for motor and cognitive actions. We want to emphasizethat the generalized complementarity principle proposedhere might play the role of coordinator. It is clear from theneurophysiological data that there exist distinct regionsresponsible for mutually exclusive behaviors such as motorand cognitive but there may also exist other regions(as pointed out by Bloedel and Bracha'I'"), which areresponsible for both kind of behaviors. We have seen fromour framework for unsharp observables and P- and R-Hlteralgebra that the distinct regions indicate the existence ofincompatible observables, and the other regions, as indicatedby Bloedel and Bracha, (19) show the possibility of jointmeasurements of incompatible observables. This might be aclear indication of probabilistic complementarity as we sawabove. Future neurophysiological data taken from patientsregarding such regions will be very helpful in verifying theconsistency of our framework. Here the selective measure-ments involved in the percept and response domains have astrong similarity with the concept of quantum filters used inquantum mechanics. Some regions of dentate nucleus in thecerebellum may be responsible for this kind of selectivemeasurement. The problem arisesregardmg the integrationprocess so as to coordinate other dentate regions of the cere-

bellum that are mainly responsible for motor actions. In thiscase,more refined experiments in neurophysiology should beperformed so as to shed new light on our understanding ofthe cognition process and responses. The complexity involvedin the human brain is such that it has not been subjected to thelevel of detailed mathematical description required for fullscientificknowledge.Wewant to emphasize that even if spedficquantum-mechanical processes are not directly involved inexplaining the cognitionprocess, quantum-like concepts suchas complementarity, quantum filters, and nonlocality may playsignificant roles in a generalized approach. As such,complementarity may extend beyond the atomic realm andmay be powerful means to understand phenomena such asthose involved in cognition, as envisaged by Bohr. More andmore refined experiments in neurophysiology will be needednot only to help us understand the cognition process itself butalso to demonstrate the viabilityof such a generalized approachas proposed here.

We finally mention another important aspect regardingusing the concept of quantum filters for selective measure-ments. Normally, in describing motor action, we needtopredict a kind of displacement, i.e., some hand or limbmovement or some eyemovement. Here the measurement isreduced to measuring displacements. In most of the experi-ments in quantum physics, a measurement normally refersto measuring displacement such as the movement of aneedle, ete. In the cognition domain, however, we do notknow exactly what type ofmeasurement is being performed.We do not even know whether we can describe those eventsin terms of physical measurements at all. But in the perceptdomain, we may think of a physical measurement as beingneurophysiological, similar to the absorption process in someof the quantum experiments. More neurophysiologicalexperiments are required to clarify these issues.

AcknowledgmentOne of the authors (SR) is indebted to Prof. Goesta H.

Granlund, Iinkoping University, Sweden, for his valuableopinions and encouragement for this work.

Received 5 June 1998.

ResumeOn croit generalement que lafonction du cervelet est d' aider le cerveau a coordonner les mouvements,cependant, des pratiques neurologiques recentes contestent ce dogme. En plus d'hre considere commeune bofte a contriile tres specialisee, le cervelet participe a plusieurs activites du cerveau incZuant la

fonction cognitive. Dans ce travail, nous examinons /'accord entre les domaines de la reponse et dela perception. On propose qu'il existe un principe similaire au ptincipe de complementarite enmecanique quantique qui fonctionne dans les domaines de reponse et de perception.

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SisirRoyCenter for Earth Observing and Space Research, SCSandPhysics and Applied Mathematics UnitIndian Statistical InstituteCalcutta 700035 India

e-mail: sroy@sdence.gmu.edu

Menas KafatosCenter for Earth Observing and Space Research, SCSandDepartment of PhysicsGeorge Mason UniversityFairfax, Virginia 22030-4444 USA

e-mail: mkafatos@compton.gmu.edu

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