{"title": "A Method for the Design of Stable Lateral Inhibition Networks that is Robust in the Presence of Circuit Parasitics", "book": "Neural Information Processing Systems", "page_first": 860, "page_last": 868, "abstract": null, "full_text": "860 \n\nA METHOD FOR THE DESIGN OF STABLE LATERAL INHIBITION \n\nNETWORKS THAT IS ROBUST IN THE PRESENCE \n\nOF CIRCUIT PARASITICS \n\nJ.L. WYATT, Jr and D.L. STANDLEY \n\nDepartment of Electrical Engineering and Computer Science \n\nMassachusetts Institute of Technology \n\nCambridge, Massachusetts 02139 \n\nABSTRACT \n\nIn the analog VLSI implementation of neural systems, it is \n\nsometimes convenient to build lateral inhibition networks by using \na locally connected on-chip resistive grid. A serious problem \nof unwanted spontaneous oscillation often arises with these \ncircuits and renders them unusable in practice. This paper reports \na design approach that guarantees such a system will be stable, \neven though the values of designed elements and parasitic elements \nin the resistive grid may be unknown. The method is based on a \nrigorous, somewhat novel mathematical analysis using Tellegen's \ntheorem and the idea of Popov multipliers from control theory. It \nis thoroughly practical because the criteria are local in the sense \nthat no overall analysis of the interconnected system is required, \nempirical in the sense that they involve only measurable frequency \nresponse data on the individual cells, and robust in the sense that \nunmodelled parasitic resistances and capacitances in the inter(cid:173)\nconnection network cannot affect the analysis. \n\nI. \n\nINTRODUCTION \n\nThe term \"lateral inhibition\" first arose in neurophysiology to \n\ndescribe a common form of neural circuitry in which the output of \neach neuron in some population is used to inhibit the response of \neach of its neighbors. Perhaps the best understood example is the \nhorizontal cell layer in the vertebrate retina, in which lateral \ninhibition simultaneously enhances intensity edges and acts as an \nautomatic lain control to extend the dynamic range of the retina \nas a whole. The principle has been used in the design of artificial \nneural system algorithms by Kohonen2 and others and in the electronic \ndesign of neural chips by Carver Mead et. al. 3 ,4. \n\nIn the VLSI implementation of neural systems, it is convenient \n\nto build lateral inhibition networks by using a locally connected \non-chip resistive grid. Linear resistors fabricated in, e.g., \npolysilicon, yield a very compact realization, and nonlinear \nresistive grids, made from MOS transistors, have been found useful \nfor image segmentation. 4 ,5 Networks of this type can be divided into \ntwo classes: feedback systems and feedforward-only systems. \nfeedforward case one set of amplifiers imposes signal voltages or \n\nIn the \n\n\u00a9 American Institute of Physics 1988 \n\n\f861 \n\ncurrents on the grid and another set reads out the resulting response \nfor subsequent processing, while the same amplifiers both \"write\" to \nthe grid and \"read\" from it in a feedback arrangement. Feedforward \nnetworks of this type are inherently stable, but feedback networks \nneed not be. \n\nA practical example is one of Carver Meadls retina chips3 that \n\nachieves edge enhancement by means of lateral inhibition through a \nresistive grid. Figure 1 shows a single cell in a continuous-time \nversion of this chip. Note that the capacitor voltage is affected \nboth by the local light intensity incident on that cell and by the \ncapacitor voltages on neighboring cells of identical design. Any \ncell drives its neighbors, which drive both their distant neighbors \nand the original cell in turn. Thus the necessary ingredients for \ninstability--active elements and signal feedback--are both present \nin this system, and in fact the continuous-time version oscillates \nso badly that the original design is scarcely usable in practice \nwith the lateral inhibition paths enabled. 6 Such oscillations can \n\nincident \nlight \n\nI \n\nv \nout \n\nFigure 1. This photoreceptor and signal processor Circuit, using two \nMOS transconductance amplifiers, realizes lateral inhibition by \ncommunicating with similar units through a resistive grid. \n\nreadily occur in any resistive grid circuit with active elements and \nfeedback,even when each individual cell is quite stable. Analysis \nof the conditions of instability by straightforward methods appears \nhopeless, since any repeated array contains many cells, each of \nwhich influences many others directly or indirectly and is influenced \nby them in turn, so that the number of simultaneously active feed(cid:173)\nback loops is enormous. \n\nThis paper reports a practical design approach that rigorously \nguarantees such a system will be stable. The very simplest version \nof the idea is intuitively obvious: design each individual cell so \nthat, although internally active, it acts like a passive system as \nseen from the resistive grid. \ndesign goal here is that each cellis output impedance should be a \npositive-real? function. This is sometimes not too difficult in \npractice; we will show that the original network in Fig. 1 satisfies \nthis condition in the absence of certain parasitic elements. More \nimportant, perhaps, it is a condition one can verify experimentally \n\nIn circuit theory language, the \n\n\f862 \n\nby frequency-response measurements. \n\nIt is physically apparent that a collection of cells that \n\nappear passive at their terminals will form a stable system when \ninterconnected through a passive medium such as a resistive grid. \nThe research contributions, reported here in summary form, are \ni) a demonstration that this passivity or positive-real condition \nis much stronger than we actually need and that weaker conditions, \nmore easily achieved in practice, suffice to guarantee stability of \nthe linear network model, and ii) an extension of i) to the nonlinear \ndomain that furthermore rules out large-signal oscillations under \ncertain conditions. \n\nII. FIRST-ORDER LINEAR ANALYSIS OF A SINGLE CELL \n\nWe begin with a linear analysis of an elementary model for the \n\ncircuit in Fig. 1. For an initial approximation to the output \nadmittance of the cell we simplify the topology (without loss of \nrelevant information) and use a naive'model for the transconductance \namplifiers, as shown in Fig. 2. \n\ne \n+ \n\nFigure 2. Simplified network topology and transconductance amplifier \nmodel for the circuit in Fig. 1. The capacitor in Fig. 1 has been \nabsorbed into CO2 \u2022 \n\nStraightforward calculations show that the output admittance is \n\ngiven by \n\nyes) \n\n(1) \n\nThis is a positive-real, i.e., passive, admittance since it can always \nbe realized by a network of the form shown in Fig. 3, where \nRl = (gm2+ Ro2 ) \n\n, and L = COI/gmlgm2\u00b7 \n\n-1 -1 \n\n, R2= (gmlgm2Rol) \n\n-1 \n\nAlthough the original circuit contains no inductors, the \n\nrealization has both capacitors and inductors and thus is capable \nof damped oscillations. Nonetheless, if the transamp model in \nFig. 2 were perfectly accurate, no network created by interconnecting \nsuch cells through a resistive grid (with parasitic capacitances) \ncould exhibit sustained oscillations. For element values that may \nbe typical in practice, the model in Fig. 3 has a lightly damped \nresonance around I KHz with a Q ~ 10. This disturbingly high Q \nsuggests that the cell will be highly sensitive to parasitic elements \nnot captured by the simple models in Fig. 2. Our preliminary \n\n\f863 \n\nyes) \n\nFigure 3. Passive network realization of the output admittance (eq. \n(1) of the circuit in Fig. 2. \n\nanalysis of a much more complex model extracted from a physical \ncircuit layout created in Carver Mead's laboratory indicates that \nthe output impedance will not be passive for all values of the trans(cid:173)\namp bias currents. But a definite explanation of the instability \nawaits a more careful circuit modelling effort and perhaps the design \nof an on-chip impedance measuring instrument. \n\nIII. POSITIVE-REAL FUNCTIONS, e-POSITlVE FUNCTIONS, AND \n\nSTABILITY OF LINEAR NETWORK MODELS \n\nIn the following discussion s = cr+jw is a complex variable, \nH(s) is a rational function (ratio of polynomials) in s with real \ncoefficients, and we assume for simplicity that H(s) has no pure \nimaginary poles. The term closed right halE plane refers to the set \nof complex numbers s with Re{s} > o. \n\nDef. I \n\nThe function H(s) is said to be positive-real if a) it has no \n\npoles in the right half plane and b) Re{H(jw)} ~ 0 for all w. \n\nIf we know at the outset that H(s) has no right half plane poles, \n\nthen Def. I reduces to a simple graphical criterion: H1s} is positive(cid:173)\nreal if and only if the Nyquist diagram of H(s) (i.e. the plot of \nH(jW) for w ~ 0, as in Fig. 4) lies entirely in the closed right half \nplane. \n\nNote that positive-real functions are necessarily stable since \n\nthey have no right half plane poles, but stable functions are not \nnecessarily positive-real, as Example 1 will show. \n\nA deep link between positive real functions, physical networks \n\nand passivity is established by the classical result7 in linear \ncircuit theory which states that H(s) is positive-real if and only if \nit is possible to synthesize a 2-terminal network of positive linear \nresistors, capacitors, inductors and ideal transformers that has H(s) \nas its driving-point impedance or admittance. \n\n\f864 \n\nOef. 2 \n\nThe function H(s) is said to be a-positive for a particular value \n\nof e(e ~ 0, e ~ ~), if a) H{s) has no poles in the right half plane, \nand b) the Nyquist plot of H(s) lies strictly to the right of the \nstraight line passing through the origin at an angle a to the real \npositive axis. \n\nNote that every a-positive function is stable and any function \n\nthat is e-positive with e = ~/2 is necessarily positive-real. \n\nI {G(jw)} \nm \n\nRe{G(jw) } \n\nFigure 4. Nyquist diagram for a fUnction that is a-positive but \nnot positive-real. \n\nExample 1 \n\nThe function \n\n(s+l) (s+40) \nG (s) = (s+5) (s+6) (s+7) \n\n(2) \n\nis a-positive (for any e between about 18\u00b0 and 68\u00b0) and stable, but it \nis not positive-real since its Nyquist diagram, shown in Fig. 4, \ncrosses into the left half plane. \n\nThe importance of e-positive functions lies in the following \n\nobservations: 1) an interconnection of passive linear resistors and \ncapacitors and cells with stable linear impedances can result in an \nunstable network, b) such an instability cannot result if the \nimpedances are also positive-real, c) a-positive impedances form a \nlarger class than positive-real ones and hence a-positivity is a less \ndemanding synthesis goal, and d) Theorem 1 below shows that such an \ninstability cannot result if the impedances are a-positive, even if \nthey are not positive-real. \n\nTheorem 1 \n\nConsider a linear network of arbitrary topology, consisting of \n\nany number of passive 2-terminal resistors and capacitors of arbitrary \nvalue driven by any number of active cells. If the output impedances \n\n\f865 \n\n'II\" \nof all the active cells are a-positive for some common a, 0O such that \nRe{(l+jwr) H(jw)} ~ 0 for all w. \n\nNote that positive real functions satisfy the Popov criterion \nwith r=O. And the reader can easily verify that G(s) in Exam~le I \nsatisfies the Popov criterion for a range of values of r. The important \neffect of the term (l+jwr) in Def. 3 is to rotate the Nyquist plot \ncounterclockwise by progressively greater amounts up to 90\u00b0 as w \nincreases. \n\nTheorem 2 \n\nConsider a network consisting of nonlinear 2-terminal resistors \n\nand capacitors, and cells with linear output impedances ~(s). Suppose \n\ni) the resistor curves are characterized by continuously \ndiffefentiable functions i k = gk(vk ) where gk(O) = 0 \nand \no < gk(vk ) < G < 00 for all values of k and vk' \nii) the capacitors are characterized by i k = Ck(Vk)~k with \no < CI < Ck(vk ) < C2 < 00 for all values of k and vk' \niii) the impedances Zk(s) have no poles in the closed right \nhalf plane and all satisfy the Popov criterion for some common \nvalue of r. \n\nIf these conditions are satisfied, then the network is stable in the \nsense that, for any initial condition, \n\nfoo( \no all branches \n\nI \n\ni~(t) dt \n\n) \n\n< 00 \n\n\u2022 \n\n(4) \n\nThe proof, based on Tellegen's theorem, is rather involved. It \n\nwill be omitted here and will appear elsewhere. \n\n\f867 \n\nACKNOWLEDGEMENT \n\nWe sincerely thank Professor Carver Mead of Cal Tech for \n\nenthusiastically supporting this work and for making it possible for \nus to present an early report on it in this conference proceedings. \nThis work was supportedJ::\u00a5 Defense Advanced Research Projects Agency \n(DoD), through the Office of Naval Research under ARPA Order No. \n3872, Contract No. N00014-80-C-0622 and Defense Advanced Research \nProjects Agency (DARPA) Contract No. N00014-87-R-0825. \n\nREFERENCES \n\n1. F.S. Werblin, \"The Control of Sensitivity on the Retina,\" \n\nScientific American, Vol. 228, no. 1, Jan. 1983, pp. 70-79. \n\n2. T. Kohonen, Self-Organization and Associative Memory, (vol. 8 in \n\nthe Springer Series in Information Sciences), Springer Verlag, \nNew York, 1984. \n\n3. M.A. Sivilotti, M.A. Mahowald, and C.A. Mead, \"Real Time Visual \n\nComputations Using Analog CMOS processing Arrays,\" Advanced \nResearch in VLSI - Proceedings of the 1987 Stanford Conference, \nP. Losleben, ed., MIT Press, 1987, pp. 295-312. \n\n4. C.A. Mead, Analog VLSI and Neural Systems, Addison-Wesley, to \n\nappear in 1988. \n\n5. J. Hutchinson, C. Koch, J. Luo and C. Mead, \"Computing Motion \nUsing Analog and Binary Resistive Networks,\" submitted to IEEE \nTransactions on Computers, August 1987. \n\n6. M. Mahowald, personal communication. \n7. B.D.O. Anderson and S. Vongpanitlerd, Network Analysis and \n\nsynthesis - A Modern Systems Theory Approach, Prentice-Hall, \nEnglewood Cliffs, NJ., 1973. \n\n8. L.V. Ahlfors, Complex Analysis, McGraw-Hill, New York, 1966, \n\np. 164. \n\n9. P. penfield, Jr., R. Spence, and S. Duinker, Tellegen's Theorem \n\nand Electrical Networks, MIT Press, Cambridge, MA,1970. \n\n10. M. Vidyasagar, Nonlinear Systems Analysis, Prentice-Hall, \n\nEnglewood Cliffs, NJ, 1970, pp. 211-217. \n\n\f\f", "award": [], "sourceid": 54, "authors": [{"given_name": "John", "family_name": "Wyatt", "institution": null}, {"given_name": "D.", "family_name": "Standley", "institution": null}]}