abraham boyarsky- towards a theory of mind

8/3/2019 Abraham Boyarsky- Towards a Theory of Mind

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1. INTRODUCTION

ones

of Mind

early

I


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2

of

may

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that jJ.manifests awareness' f the brain and revealsthe underlying brain dynamical system (X, T). TnSection4 we use he structure of the Markov modelto estimate nformation rates or the consciouspartof the brain. In Section 5 we discllss how thedynamical system model can be used to create auniversal model in which brains are connected. nSection6 we show that the dynamical systemmodelhas a connection to q Iantum theory in view of thefact that wave unctions, when squared,give rise toprobability density functions that are the densitiesof SRB measures for the spatia.! processes ofquantum mechanics.

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2. DYNAMICAL SYSTEM NOTATIONLet X be a space i.e., a collectionof points) andT: X -- X a transfonnation. When we want tomeasuresubsetsof X, we first classify the subsetsto be considered. These are usually the Borelsubsets,a Borel subsetbeing the smallestcollectionof subsetsof X containing the open subsets unionsof balls) and is closed under taking complementsand umons of countable many members of thiscollection. Thesesubsetsare said to be measumble.A function which associates o each Borel set .\positive number in such a way that for eachcountable collection of disjoint sets Ai> one has~(UAi) = 2:~(Ai) is called a measure. It is aprobability measure f Jk(X) = I. We consider 0111yprobability measures. By the support of a mea-sure we me-an he smallest (closed) Borel subset Ssuch that ~(S) = 1.We say fJ, s supported on S.Furthermore, S s an invariant set which means hatT(S)


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In-I.Jim - LXA(T'x) =peA)n-oo n to.()

are

1n-IHrn -~{j-oonL-J Tlx = J1.

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f.L cts on subsetsof X, that is on space atherthan in time and, as such, is always 'avail-able' as a source of knowledge about thedynamical system. Furthermore, Jl can beused o reveal T.Jl actsas a global attractor as evidenced y (2),i.e., for any starting point (in a large set ofstarting points), the time-averagedorbit con-verges weakly) to f.L.

an

weakly, (2)

if 0 $ x < t.if! < x $ I,hown in Fig. 2.l the partition P of X consist of PI == 0,1) andi!, 1]. Then Tis Markov for this partition sinceI=PIUP2 and 1T.P2)=PIUP2' Obviously,Into as it maps each element of the partition

a


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4

nudge is sufficient to expose he system's nherentchaotic behavio\lr and the new orbit will settlequickly into the system'sSRB measureA.To summarize: for most starting points x, theSRB measure which is known to exist) will comeinto actual being via the averaging process n (3).Due to the instability of periodic orbits (T isexpanding), any cycle is eventually derailed to apoint whoseorbit is chaotic and the associated rbitof Dirac measures quickly settles into the SRBmeasure. The starting points that lead to chaoticorbits have Lebesgue measure I and hence areubiquitously available to settle the dynamics intothe SRB measureA,Once the SRB measure s known to exist, t is thenow of time and the time-averagingmechanism hatdisplays he 8RB measure.This averagingmechan-ism is nothing other than a kind of check-up of allthe components of the brain system, hat they arefunctioning properly and available for service.We will use hese udimcntay notions n the sequelto study brain dynamics. To prepare for theapplication in a higher dimensional setting, notethat A possesseshe following properties,Awareness Let x. be any point in X which isnot a periodic point and let A. be a very small setcontaining x", Since T is expanding, the orbit{Tix*}~:o! movesquickly out of A", say n m < 1Zsteps.Thus,

-.

)./14' ~ (m/n)fi = Ct)_\i, (4)which is the (scaled)Dirac measureat x.. Knowingthis measure s the sameas knowing x". Hence A j!;'aware' of x'.Inversion Since X and the partition Pareknown, knowledge of A det~nnines (almost uni-quely) the transformation T. This follows froma simple application of the matrix form of theFrobenius-Operator equation [2] for piecewiselinear transformations.In summary, he SRBmeasure).has he ollowingproperties:1. >. s 'aware' of almost all points in its support asevinced by (4).


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6

brain

4. INFORMATION RATES

Pi> O. Hencedynamical system s given byq

LPilog2Pi.;=1H(P,1) =-

and

given by [13, Theorem 2.50]

(6)Hence, the maximum information rate for this q-

parallel system is q log2q. For q = 10, he maximuminformation rate is approximately 33 bitsfs whichaccordswell with the estimate f between 5 and50bitsfs given n [9, p. 60] by meansof a com-pletely different line of reasoning.

by

5. UNIVERSAL MODEL OF MINDLet us now consider N brains {B" B2,..., BN}functioning at the same time. Since these brainsare independent of each other we can displaytheir dynamics on the space of images X =XI U X2 u. . U X N, where Xi denotes all the brainimageson Bi. Let T[: Xi--.>Xi be he image transfor-mation on Xi Then T,{Xj) ~ Xi, i.e., each space ofimagesXi is invariant under Tj. In Fig. 4 we showthis multi-brain situation symbolicaHy.

By changing Ti slightly we can obtain an inter-connectedbrain system.Let ussuppose hat for eachbrain Xi, Tj is altered to Ti- as follows: on a verysmallsetljinXi, we et 1';(1i) = X, that is, Ticarries

and

a

1XIf0.8 .

, .o. . 4..0.4 .XI

0 X, 0.2 ~ 0.4 0.8 ')( II 1.6FIGURE 4 Independent brains.


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asmalbrain.as is st

Forexists jwhen rJ1.i, the ~in X"nonethsmall cthroug!SRBmon ailFinallyextend~other b

6. corME4

In quatwhen scfunctiolThen frdiffusiogenera i

7or random map [8] whoseby '(x), i.e.,

J.tt(A)= J,(x)dx.7.(i)

'BSERVATIONS[)cadonof Mind


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folJowing symbolic equation:

(iii) Relative Consciousness

IS

(iv) Timelessness f Mind

of piecesof the physical spaceD, Sand f-Lare boundby the very existenceof f-L.Even if the underlyingspaceB disintegrates, 1-ontinues to exist, even fmerely as a mathematical entity. Thus, mind - oncecreated by an interesting transformation and theflow oftime - exists orever after.

Refe,oences[1] P. Billingsley, Ergodic Theory {/lid Information, R.E. Krieger

Publishing Co., 1978.[2J A. BoyarSkyand P. Gora, Laws of Chao x, Birkhauser ]997.[3] A. Boyarsky, P. Gora and V. Lyabumov, Dynamics onspaces of compact subsets with application to brainmodeling,Jour. Math. Anal. Appl. 216, 569-580,1997.[4] F. Crick and C. Koch, The problem of consciousness,Scientific American, Sept. 1992.[5J F. Crick, The A.f{0I1i.5hingHypothesis, Tlte Sd'!Iltific Searchfor the Sou!, Scriibners, N.Y., 1994.[6} R. Devaney, All Introductiol ilO Cltaotic Dynamical System.r,2nd edition, Addison Wesley, 1989.[7J P. Gora and A. Boyarsky, Absolutely continuous invariantmeasures fo piecewise expanding C2 transformations in R",Israel.lour. Math., 67(3),272- 286, 1989.[8} P. Gora and A. Boyarsky, A lattice spacetime for the slitexperiments of quantum n1echanics, Physics Letters A 236,263--269,1997.[9] N. Herbert, Elemental Mind, Dutton, 1993.[10] J. Horgan, Can science explain consciousness? ScientifiCAmerican, July 1994.[IIJ D.G. Kendall and E.F. Harding, Stoclta.rtic Analyst!', Johl)Wiley,N.V., 1974.[l2J R. Mane, Ergodic Theory of Differentiable Dynamic.~,Springer-Verlag, N.Y., 1987.[13] N. Martin and J. England, Mathe/J1atil'l/!Theory o.lEmropy,Addison-Wesley, 1981.[14] W. de Melo and S. van Strien, One-dimensional dynamics,Springer-Verlag 1993.[15] E. Nelson, Dynamical Theory of Brownian Motion, PrincetonUniversity Press, ]967.[16J R. Penrose, Thl! Einpo/'er's Nell' Mind, Oxford UniversityPress, 1989.[17] J. Serra, Image Analysis and Mathematical Mo/'phology,Academic Press, N.Y., 1982.

abraham boyarsky- towards a theory of mind

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