Contains reports on two research projects.National Institutes of Health (Grant 5 ROl NB-04985-04)U. S. Air Force (Aerospace Medical Division) under Contract AF33(615)-3885DSR Project 55-257Bioscience Division of National Aeronautics and Space Administration through Contract NSR 22-009-13

da Fonseca, J. L. S.

McCulloch, Warren S.

Moreno-Diaz, R.

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XXIII. NEUROPHYSIOLOGYAcademic and Research StaffDr. W. S. McCulloch Dr. T. O. Binford Dr. A. NatapoffProf. J. Y. Lettvin Dr. S-H. Chung Dr. S. A. PapertProf. P. D. Wall Dr. J. L. S. da Fonseca Dr. A. TaubProf. M. Blum Dr. H. Hartman B. HowlandProf. J. E. Brown Dr. Dora Jassik-Gerschenfeld Diane MajorProf. P. R. Gross Dr. K. Kornacker W. H. PittsProf. B. Pomeranz Dr. T. McLardy Sylvia G. RabinDr. R. Moreno-DiazGraduate StudentsE. E. Fetz J. I. SimpsonG. P. Nelson W. A. WrightA. CONTRAST DETECTORStLateral substractive inhibition may occur at the retinal bipolar level, therebyenabling a contrast-detection process in which each "contrast bipolar" cell could detectany spatial change of the intensity of light incident in its receptive field. In such a sit-uation, the locus of the contrast bipolar cells that fire at any time gives the contour ofany sharply contrasted image on the retina at this time. A contrast-detector ganglioncell receiving signals from an area that contains many contrast bipolar cells therebysenses a significant part of that contour as limited by the object size-ganglion fieldrelationship. Its rate of firing is different for different shapes and sizes of the imageand for different velocities of a moving image. It is, therefore, desirable to obtainsimple expressions relating ganglion tone to the aforementioned parameters that areconsistent with, and supported by, neurophysiological evidence.In particular, we shall be concerned with the dependence of ganglion tone upon thelength of the contrast, or edge, of the image and upon its velocity. By applying a pro-cess used to model a specific visual ganglion cell,l we can obtain one expression forsuch a dependence. The process to which we refer is the lateral (nonlinear) inhibition atthe level of the ganglion cell dendrites. All that we require is, first, that there be asingle horizontal dendritic layer for the contrast ganglion cell, and second, that signalsarriving there from bipolar cells interact by lateral nonlinear inhibition. Either divisiveinhibition of exponential inhibition would be adequate for this task. We chose, for sim-plicity, to use exponential inhibition 2 in our formulations.This work was supported by the National Institutes of Health (Grant 5 ROlNB-04985-04), and in part by the U. S. Air Force (Aerospace Medical Division) underContract AF33(615)-3885.tThis work was done partly at the Instrumentation Laboratory under the auspices ofDSR Project 55-257, sponsored by the Bioscience Division of National Aeronautics andSpace Administration through Contract NSR 22-009-138.QPR No. 85 321(XXIII. NEUROPHYSIOLOGY)For the sake of simplicity, we assume: First, that the contrast bipolars are uni-formly distributed throughout the retina and that their number is large enough to per-mit the assignment of "linear density" (X), the number of contrast bipolar cells per unitretinal length, as a constant. Second, we assume that bipolar cells give rise to pulses(action potentials) of constant amplitude (p) and constant duration (T), and that they con-tain refractory periods of sorts.If the image of a sharply contrasted object is stationary with respect to the retina,and 1 is the length of the edge in the contrast ganglion cell receptive field, this cellreceives pulses from n o contrast bipolar cells, in whichn o = l. (1)If the image moves with respect to the retina, with velocity of absolute value u, theganglion cell will receive pulses from ne additional bipolar cells such that ne is approx-imately given byn e = u - Tl A. (2)The total number, n, of bipolar cells exciting the ganglion cell is given byn = n + ne = lX(l+uTX). (3)If the lateral exponential inhibition occurs, the contribution of a single bipolar cellto the activity of the ganglion cell is given byp . e-kp(n- 1), (4)where k is a constant that determines the strength of the inhibitory action. Under theassumption that contributions are additive, the total activity, Ac, of the ganglion cellis given byAc = p e-kp(n-l), (5)where the summation is over all of the bipolar cells that are firing, that is, n. Then,Ac =npe-kp(n 1) (6)By substituting Eq. 3 in Eq. 6 and renaming the constants, we obtainAc = K 11(1+K 3 u) e-K 1(1+K 3 (7)whereQPR No. 85 322(XXIII. NEUROPHYSIOLOGY)K 1 = p ekpK2 = k p XK3 = T X.Let the frequency (tone), f, of the ganglion cell be linearly related to its activity;that is,f = aAc + b, f >0, (8)where b, if positive, is the spontaneous firing frequency. A negative value of b maybe interpreted as threshold.Let us define z asz - 1(1+K 3 u). (9)afThe maximum value of f occurs when = 0, that is, for a value of z (representedby Zopt), in whichz = 1 (10)opt K2For a stationary object (u = 0) there is an optimum edge length in the retinal fieldwhich produces a maximum response. This is1 = 1 (11)opt K2For moving objects of constant edge length, 1 o, there is an optimum velocity, Uop t 'which produces a maximum f and is given byopt K 3  10 ) (12)In any case, the maximum frequency is given byKfmax =a K + b. (13)Since only the absolute value of the velocity u appears in the expressions above, anegative value for Uopt has no meaning. Therefore, for a given edge length, 1o , themaximum frequency can be obtained only ifKz1 < 1. (14)The case K2 1 = 1 corresponds to the optimum stationary edge, that is, a maximumQPR No. 85 323(XXIII. NEUROPHYSIOLOGY)ganglion tone that is due to a stationary edge, as given by Eq. 11.Properties that are qualitatively similar to the ones described here - and that haveoriented our work- have been reported for the group 1 ganglion cells in the frog'sretina.3, 4R. Moreno-DiazReferences1. R. Moreno-Diaz, "Modeling the Group 2 Ganglion Cell of the Frog's Retina,"Quarterly Progress Report No. 81, Research Laboratory of Electronics, M. I. T. ,April 15, 1966, pp. 227-236.2. Exponential inhibition may be formulated as follows: Let E and I be the excitatoryand inhibitory signals. After inhibition, the resulting signal (activity) is E * eKIwhere K is a constant. See, for example, J. J. Shypperheyn, "Contrast Detectionin Frog's Retina," Acta Physiol. Pharmacol. Neerl. 13, 231-277 (1965).3. J. Y. Lettvin, H. R. Maturana, W. H. Pitts, and W. S. McCulloch, "Two Remarkson the Visual System of the Frog," in Sensory Communication, W. A. Rosenblith(ed.) (The M. I. T. Press, Cambridge, Mass., and John Wiley and Sons, Inc.,New York and London, 1961), pp. 757-776.4. R. Grusser-Cornehls, O. J. Grusser, and T. H. Bullock, "Unit Responses in theFrog's Tectum to Moving and Non-moving Visual Stimuli," Science 141, 820-822(1963).QPR No. 85 324(XXIII. NEUROPHYSIOLOGY)B. INSIGHT INTO NEURONAL CLOSED LOOPS FROM SHIFT-REGISTER THEORY*In 1943, Pitts and McCulloch I produced three theorems showing that nets with circlescould compute only those numbers that a Turing machine could compute with a finitetape. Recently, Minsky and Papert 2 have reported several theorems applicable to them,3-6and our group has presented some in previous reports. Since Wiener's pioneeringtreatment of nonlinear filters, little progress has been made, until last year whenCaianiello published two items in Kybernetik. 7' 8In the meantime, a theory of shift registers has grown up independently, dependingon Galois Fields, initiated by Huffman's 9 ' 10 analysis of their use in coding, and a sub-11-24sequent evolution of theorems of limited generality. It has not been recognizedthat all such devices were within the scope of nerve net theory, although the converseis not generally true in the present state of the art. As their theory is relatively com-plete 2 5 - 9 (except for fully nonlinear shift registers with inputs), it is clear that shift-register theory can sometimes be extended from Boolian functions of 0 and 1 to any30-32finite field of integers, and in some cases to the field of real numbers, and perhapseven of complex numbers.There is no apparent reason why these theories may not ultimately be extended toall closed-loop structures in finite automata of neuron nets of proper neuromimes. Westill do not know how to extend this theory to the general problems of nonlinear anasto-motic nets with inputs.(a) (b)Fig. XXIII-1. Nonlinear feedback shift register for control (a) and command (b).Even without that, the utility of these notions foreshadows a great extension of neu-ronal modelling in the design of circuits for command and control, as well as in the pro-gramming of commands for which shift registers are eminently suitable. We alreadyhave simple solutions for temporal sequences in nonlinear feedback shift registers.*This work was done partly at the Instrumentation Laboratory under the auspices ofDSR Project 55-257, sponsored by the Bioscience Division of the National Aeronauticsand Space Administration through Contract NSR 22-009-138.QPR No. 85 325(XXIII. NEUROPHYSIOLOGY)For example, we have found conditions for driving the state of nonlinear feedbackshift registers with input through any logical gate of types (a) and (b) (see Fig. XXIII-1).In this figureI = InputL = Logical gateThe vector that defines the state of the n delay elements of an n th-order shiftregisterf(X) = Any Boolian function of the n components of Xg(X, I) = Any Boolian function of the components of X and I.W. S. McCulloch, J. L. S. da FonsecaReferences1. W. S. McCulloch and W. H. Pitts, "A Logical Calculus of the Ideas Immanent inNervous Activity," Bull. Math. Biophys. 9, 127-247 (1943).2. M. L. Minsky and S. A. Papert, "Unrecognizable Sets of Numbers," Project MACProgress Report III, Massachusetts Institute of Technology, July 1965-July 1966,pp. 18-20.3. M. Blum, "Properties of a Neuron with Many Inputs," in Principles of Self-Organization, H. von Foerster and G. Zopf (eds.) (Pergamon Press, Oxford andNew York, 1961), pp. 95-119.4. C. P. J. 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, 1965, p. 253.5. R. Moreno-Diaz, "Realizability of a Neural Network Capable of All Possible Modesof Oscillation," Quarterly Progress Report No. 82, Research Laboratory of Elec-tronics, M. I. T., July 15, 1966, pp. 280-285.6. R. Moreno-Diaz, "Stability of Networks with Loops," Quarterly Progress ReportNo. 83, Research Laboratory of Electronics, M.I.T., October 15, 1966,pp. 165-171.7. E. R. Caianiello and A. De Luca, "Decision Equation for Binary Systems. Appli-cation to Neuronal Behavior," Kybernetik 3, 33-40 (1966).8. E. R. Caianiello, "Decision Equation and Reverberation," Kybernetik 3, 98-99(1966).9. D. A. Huffman, "The Synthesis of Linear Sequential Coding Networks," in Infor-mation Theory, C. Cherry (ed.) (Butterworths, London, 1955), pp. 77-95.10. D. A. Huffman, "A Linear Circuit Viewpoint on Error-Correcting Codes," IRETrans. , Vol. IT-2, 20-28, 1956.11. M. Cohn, "Controllability in Linear Sequential Networks," IRE Trans., Vol. CT-9,pp. 74-78, 1962.12. B. Elspas, "The Theory of Autonomous Linear Sequential Networks," IRE Trans.,Vol. CT-6, pp. 45-60, 1959.13. P. Fire, "Boolean Operations on Binary Markov Chains," Report EDL-L 27,Sylvania Electronic Defense Laboratories, 1964.14. B. Friedland and T. E. Stern, "On Periodicity of States in Linear Modular Sequen-tial Circuits," IRE Trans., Vol. IT-5, pp. 136-137, 1959.QPR No. 85 326(XXIII. NEUROPHYSIOLOGY)15. B. Friedland, "Linear Modular Sequential Circuits," IRE Trans., Vol. CT-6,pp. 61-68, 1959.16. W. Golomb, "Sequences with Randomness Properties," Final Report on ContractNo. W 36-039 SC-54-36611, Glenn L. Martin Co., Baltimore, Md., 1955.17. J. Hartmanis, "Linear Multivalued Sequential Coding Networks," IRE Trans.,Vol. CT-6, pp. 69-74, 1959.18. J. Massey, "Shift-Register Synthesis and BCH Decoding" (personal communication).19. F. J. Mowle, "Enumeration and Classification of Stable Feedback Shift Registers,"Technical Report No. EE-661, Department of Electrical Engineering, Universityof Notre Dame, Notre Dame, Indiana, 1966.20. W. W. Peterson, Error Correcting Codes (The M. I. T. Press, Cambridge, Mass.,1961).21. C. V. Srinivasan, "State Diagram of Linear Sequential Machines," J. FranklinInst. 273, 383-418 (1962).22. T. E. Stern and B. Friedland, "The Linear Modular Sequential Circuit Generalized,"IRE Trans., Vol. CT-8, pp. 79-80, 1961.23. N. Zierler, "Several Binary-Sequence Generators," Technical Report No. 95,Lincoln Laboratory, M. I. T., 1955.24. N. Zierler, "Linear Recurring Sequences," SIAM J. 7, 31-48 (1959).25. S. W. Golomb, "Non-Linear Shift Register Sequences," Memorandum No. 20-149,Jet Propulsion Laboratory, California Institute of Technology, Pasadena, Cali-fornia, 1957.26. S. W. Golomb, L. R. Welch, and R. M. Goldstein, "Cycles from Non-Linear ShiftRegisters," Jet Propulsion Laboratory, California Institute of Technology, Pasadena,California, 1959.27. K. B. Magleby, "The Synthesis of Nonlinear Feedback Shift Registers," TechnicalReport No. 6207-1, Stanford Electronics Laboratories, Stanford, California, 1963.28. J. Massey and R. Liu, "Equivalence of Nonlinear Shift Registers," IEEE Trans.,Vol. IT-10, pp. 378-379, 1964.29. K. Fukunaga, "A Theory of Nonlinear Autonomous Sequential Nets Using Z Trans-forms," IEEE Trans. , Vol. EC-13, pp. 310-312, 1964.30. T. Kubista and J. Massey, "Pseudo-Linear Generalized Shift-Registers," Depart-ment of Electrical Engineering, University of Notre Dame, Notre Dame, Indiana,1966.31. J. Massey and R. Liu, "Application of Lyapunov's Direct Method to the Error-Propagation Effect in Convolutional Codes," IEEE Trans. , Vol. IT-10, pp. 248-250,1964.32. C. M. Rader and B. Gold, "Digital Filter Design Techniques in the FrequencyDomain," Proc. IEEE 55, 149-170 (1967).QPR No. 85 327

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