Contains reports on two research projects.National Institutes of Health (Grant NB-4897-03)U.S. Air Force (Aerospace Medical Division) under Contract AF33(615)-3885The Teagle Foundation, Inc.Bioscience Division of National Aeronautics and Space Administration, Contract NSR 22-009-13

Kilmer, W. L.

McCulloch, Warren S.

Moreno-Diaz, R.

DSpace@MIT

XXII. NEUROPHYSIOLOGYAcademic and Research StaffDr. W. S. McCulloch Dr. K. Kornacher Dr. T. G. Smith, Jr.Dr. J. Y. Lettvin Dr. R. Moreno-Diaz Dr. A. TaubProf. P. D. Wall Dr. T. McLardy B. HowlandProf. M. Blum Dr. A. Natapoff Diane MajorProf. J. E. Brown Dr. S. A. Papert W. H. PittsDr. H. Hartman Dr. Barbara C. G. Pickard Sylvia G. RabinDr. W. F. PickardGraduate StudentsE. E. Fetz J. I. Simpson Barbara G. WickelgrenL. M. Mendell W. A. WrightA. SUMMARY OF RESEARCH PROGRESS: THEORY OF THE RETICULARFORMATIONResearch on the functional organization of the reticular core of vertebrate controlnervous systems has reached a significant landmark. Our problem is as described inQuarterly Progress Report No. 76 (page 313), but our progress has been achieved witha somewhat different model than that mentioned there.The new model is enclosed within heavy lines in Fig. XXII-1, with everything outsideonly for generating an appropriately structured environment for computer simulation.The yij in Fig. XXII-1 are all 3-variable symmetric switching functions of the binary a-..The typical module interconnection scheme is suggested by the M 5 hookup. Each modulein Fig. XXII-1 is a hybrid probability computer, with schematic as shown in Fig. XXII-2.It receives 4-component probability vectors Pgi from above and Pai from below, as wellas generating a corresponding Pri from its Nia part. The jth component in each case isthe probability as computed by the module of origin that the over-all reticular formationmodel's present yij input-signal configuration is properly a mode j one. The Pai' Psi'and P .vectors are passed componentwise through an f function as shown in Fig. XXII-3,and weighted in the subsequent 'Av' units according to formulas of the typeCP +CP +CPr Tr aa 66rT a 6where C = C C Q, with all factors variable and determined according to two moduledecoupling principles and a potential command principle which demands that informationThis work was supported by the National Institutes of Health (Grant NB-4897-03),the U.S. Air Force (Aerospace Medical Division) under Contract AF33(615)-3885, andby a grant from The Teagle Foundation, Inc.QPR No. 82 275( - ) OY23 Y26 30S4.25 . 8  7.5 2 6i .....-. ... ........S5BY31 I I 11 '35Fig. XXII-l. Simulation model (S-Retic).QPR No. 82 276Fig. XXII-2. A typical M i of Fig. XXII-1.f(p)2.80.30.2- 0.40.2 0.3 1.0 PFig. XXII-3. The f(p) function.QPR No. 82 278P23W14p5MI 10, 1M6LINEAR THRESHOLDLOGIC ELEMENTPROGRAMMED SEQUENCEOF OUTPUT SIGNALS TORELEASE EFFECTOR PATTERNw10 ,4p7(M7)P9czzM8Fig. XXII-4. S-Retic output scheme.QPR No. 82 279(XXII. NEUROPHYSIOLOGY)constitute authority. The h, T, h-1, N, and UD blocks in Fig. XXII-2 are to insure thatevery Pi is an appropriately normalized and delayed probability vector. Figure XXII-4shows our Fig. XXII-1 model's output model detection scheme.The model has been successfully simulated on the Honeywell Computer at Instrumen-tation Laboratory, M. I. T. , in collaboration with J. Blum, W. L. Kilmer, E. Craighill,and D. Peterson. The model converged to the correct output model indication in eachof approximately 50 test cases, and always in from 5 to 25 time steps. This is just whatwe had hoped for.We are now concentrating on the functional design of a considerably enriched modelthat can handle conditioning and extinction in a satisfactory time-domain sense. Thedesign will again be programmed for simulation on the Instrumentation Laboratory com-puter.W. S. McCulloch, W. L. Kilmer[Dr. W. L. Kilmer is now at Michigan State University, East Lansing, Michigan.]References1. W. S. McCulloch, W. L. Kilmer, J. Blum, and E. Craighill, "Toward a Theory ofthe Reticular Formation," a paper presented at the IEEE/Cybernetics Session' IEEEInternational Convention, New York, March 14, 1966.2. W. S. McCulloch, W. L. Kilmer, J. Blum, and E. Craighill, "A Cybernetic Theoryof the Reticular Formation," a paper presented at the Bionics Symposium, Dayton,Ohio, May 2-5, 1966.B. REALIZABILITY OF A NEURAL NETWORK CAPABLE OF ALL POSSIBLEMODES OF OSCILLATION1. IntroductionDr. McCulloch has called our attention to the need for investigating the modes ofoscillation of neural nets with feedback and under constant input. The question "Howmany possible modes of oscillation are there for N neurons?" has already beenanswered by C. Schnabel. 1 There are (K-1) K possible modes of oscillation.K=2The next question is, "Are all of these modes realizable with a fixed anatomy?" Theanswer is affirmative, provided there is a minimum number of input lines to the net-work. The proof is presented here.*This report was prepared at the Instrumentation Laboratory under the auspices ofDSR Project 55-257, sponsored by the Bioscience Division of National Aeronautics andSpace Administration, Contract NSR 22-009-138.QPR No. 82 280(XXII. NEUROPHYSIOLOGY)2. Y-NetworksConsider N formal neurons with interactions of afferents (those described byMcCulloch in "Agathe Tyche"2) forming a one-layer network with M binary inputs, andin which each neuron binary-output feeds back to the same neuron and to all the others.The number of inputs to each neuron is, then, M + N. At any time t the output (or state)- 0 or 1 - of any neuron is determined by the value of its inputs at time t - 1. Therefore,the state of any neuron at time t is determined by the states of all of the neurons and theinputs to the network, both at time t - 1. At any time t, the state of the network isdefined as the array of N zeros and ones that indicate the state of each neuron at thistime. Thus, we can say that the state of the network at time t is determined by thestate of the network and the input configuration, both at time t - 1.For a network of N neurons, there are 2N possible states. We can imagine someparticular networks in which some of these 2N states are never reached. We are inter-ested in those networks in which any of the 2N states may be reached from any initialstate by at least 1 input sequence. Such networks are here referred to as Y-networks.Thus, an 2 -network is a network of N neurons forming one layer in which the outputof each neuron is connected to itself and all of the others, and any of the 2N states maybe reached from any initial state by some input sequence.Consider the set of all possible states S = (Si' S2 ' .. S, SN) and the set of all possibleconfigurations of the inputs S = (W 1 ,W 2 ,... ,W2M) of an Y-network with M inputs.From these sets, we form all possible doubles(Si, Wk),where Wk E W and Si E S. We have Z 2 N = 2  doubles. We now generate 2 M+Nsuccessors of the form(SiWk) - Sjby arbitrarily assigning to each double (Si, Wk) an element Sj E S and only one, and usingM+Nany Sj E S at least once. The set of 2 successors generated in this manner isreferred to as a "set of successors, v(N,M)." We generate all possible sets v(N, M),and form from them a new ensemble /. Thus, Y' is the ensemble of all possible setsof successors v(N, M).Returning to any one Y-network, if S. is the state at any time t - 1, Wk is the input1M+Nconfiguration at time t - 1, and Sj is the state at time t, we can form a set, v, of 2 M+Nsuccessors of the formQPR No. 82 281[Si(t-1), Wk(t-1) ] - Sj(t).(XXII. NEUROPHYSIOLOGY)which describes completely the behavior of the net. Obviously v E -/. We will provethat the inverse is also true, i. e. , given any arbitrary v, there is an Y-network thatis described by v.LEMMA 1. Given an arbitrary set of successors v(N, M) E 'r, it is always possibleto design an Y-network of N neurons and M inputs that verifies v(N, M).Proof. The proof consists in generating, from v(N, M), N Venn diagrams of N + Minputs each. The network can be designed from the N Venn diagrams (see Blum 3 ).We first note that each double (Si, Wk) of each successor (Si, Wk) - Sj of v(N,M)determines one area in a Venn diagram of N + M inputs.Let us assume that we have drawn the Venn diagram for the first neuron. Then, weput a jot in those areas of the Venn represented by all (Si, Wk) for which the S indicatethat the state of that neuron is 1 (fires). We repeat the same for all of the N neurons.Thus, for each combination (Si, Wk), the corresponding Sj is determined.3. Modes of OscillationConsider a system of N formal neurons and M inputs. If we define the state of thesystem at time t as the array of N zeros and ones that indicate the state of each neuronNat time t, there are 2 possible states. A mode of oscillation of the system is definedas any sequence of states that is repeated under a constant input and involves more thanone member state. A k-dimensional mode of oscillation is a mode that passes throughk different states.The number N of possible modes of oscillation of N neurons is (see Schnabellk= 2 NNo = ( !k=2LEMMA 2. All No possible modes of oscillation of N neurons can be described bya set of successors v(N, M) such thatk=2M > log 2  L -(K-i)!/Ewhere Ek indicates the maximum whole number that is less than or equal to 2 N/k.EProof. Any k-dimensional mode of oscillation can be divided in k-steps. Each stepcan be expressed by a successor(Si, Wk) - Sj,where Si and S. are the states of the step, and Wk is the constant input word thatQPR No. 82 282(XXII. NEUROPHYSIOLOGY)produces the mode. If we require that two modes with at least one state in common can-not be specified by the same input configuration, we can insure that for each double(Si, Wk) there is one and only one S..Thus the number of k-dimensional modes of oscillation that may result from any oneinput configuration is the largest integer that is less than or equal to ZN/k. We denotethis integer by . The number of input configurations necessary to specify allk-dimensional modes, is(k-1) ! 2k)(number of k-dimensional modes divided by the number of k-dimensional modes speci-fied by each input configuration). The minimum total number of input configurationsthen isand therefore the number of input lines, M, has to be such thatk= 22 M> (k-) (2 N)k= 2 2 )EEorM g2  (k-1)! 2N)Since Z k! Kk) doubles have been used in describing all modes of oscillation,k=2k=2 2M+N h Nk= 2 2- k! doubles. This could be, for example, the state S .... tok=2all of the doubles left.QPR No. 82 283(XXII. NEUROPHYSIOLOGY)THEOREM. Given N neurons and M input lines such thatk= 2M > log 2k=2(k-i)! 2 N)( N )k \ k /2( k Eit is always possible to design an Y-network that verifies all possible modes of oscil-lation.Proof. All possible modes of oscillation of N neurons can be expressed by a setv(N, M) of successors such thatk= 2 NM > log 2k=2(k-1) ! (ZN(N N)22N (kE(Lemma 2).According to Lemma 1, it is always possible to design an Y-network that verifies anyv(N, M), in particular, that which describes all possible modes of oscillation.The minimum number of jots, J, per Venn diagram for such an Y-network is thesame for all of the neurons of the network. This number can be computed as follows.The number of modes of oscillation that pass through any one state isNk=2(k-1)! (kN_This number gives the number of doubles that correspond to the same S. in all suc-cessors (Si, Wk) -S.. In describing the first neuron, for example, we put jots in theVenn areas for which S. indicates that the neuron fires. There are 2 of these statesS.. Therefore, the number of jots, j, for the Venn of that neuron is at leastJNk=2,J/= 2 N-I (k-)and JI/ is the same for all neurons.R. Moreno-DiazReferences1. C. P. G. Schnabel, "Number of Modes of Oscillation of a Net of N Neurons," Quar-terly Progress Report No. 80, Research Laboratory of Electronics, M. I. T. , Jan-uary 15, 1966, p. 253.QPR No. 82 284(XXII. NEUROPHYSIOLOGY)2. W. S. McCulloch, Embodiments of Mind (The M.I.T. Press, Cambridge, Mass., 1965),pp. 203-215.3. M. Blum, "Properties of a Neuron with Many Inputs," Principles of Self-Organiza-tion, H. Von Foerster and R. Zopf (eds.) (Pergamon Press, Oxford, London,New York and Paris, 1962), pp. 95-119.QPR No. 82 285

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