O.97ms both periodic and chaotic attractors are found. \n\n1.0-\n\nV2 -\n\no -\n\n1.0-\n\nV2 -\n\no -\n\n't \n\nA \n\n0 \n\nVI \n\n1.0 \n\n0 \n\nVI \n\n1.0 \n\nFigure 4. Period Doubling to Chaos as the Delay in Neuron 1 is Increased. \n\nChaos in the network of Fig.4 is closely related to a well-known chaotic delay(cid:173)\ndifferential equation with a noninvertible feedback term [Mackey and Glass,1977]. The \nnoninvertible or \"mixed\" feedback necessary to produce chaos in the Mackey-Glass \nequation is achieved in the neural network - which has only monotone transfer \nfunctions - by using asymmetric connections. \n\nThis association between asymmetry and noninvertible feedback suggests that \nasymmetric connections may be necessary to produce chaotic dynamics in neural \nnetworks, even when time delay is present. This conjecture is further supported by \nconsidering the two limiting cases of zero delay and infinite delay, neither of which show \nchaotic dynamics for symmetric connections. \n\nIV. CONCLUSION AND OPEN PROBLEMS \n\nWe have considered the effects of delayed response in a continuous-time neural network. \nWe find that when the delay of each neuron exceeds a critical value sustained oscillatory \nmodes appear in a symmetric network. Stability analysis yields a design criterion for \nbuilding stable electronic neural networks, but these results can also be used to created \ndesired oscillatory modes in delay networks. For example, a variation of the Hebb rule \n[Hebb, 1949], created by simply taking the negative of a Hebb matrix, will give \nnegative real eigenvalues corresponding to programed oscillatory patterns. Analyzing the \nstorage capacities and other properties of neural networks with dynamic attractors remain \n\n\f576 \n\nMarcus and Westervelt \n\nchallenging problems [see, e.g. Gutfreund and Mezard, 1988]. \n\nIn analyzing the stability of delay systems, we have assumed that the delays and gains of \nall neurons are identical. This is quite restrictive and is certainly not justified from a \nbiological viewpoint. It would be interesting to study the effects of a wide range of \ndelays in both symmetric and non-symmetric neural networks. It is possible, for \nexample, that the coherent oscillation described above will not persist when the delays \nare widely distributed. \n\nAcknowledgements \n\nOne of us (CMM) acknowledges support as an AT&T Bell Laboratories Scholar. \nResearch supported in part by JSEP contract NOOOI4-84-K-0465. \n\nReferences \n\nS. Amari, 1971, Proc. IEEE, 59, 35. \nS. Amari, 1972, IEEE Trans. SMC-2, 643. \nU. an der Heiden, 1980, Analysis of Neural Networks, Vol. 35 of Lecture Notes in \n\nBiomathematics (Springer, New York). \n\nK.L. Babcock and R.M. Westervelt, 1987, Physica 28D, 305. \nM.A. Cohen and S. Grossberg, 1983, IEEE Trans. SMC-13, 815. \nA.H. Cohen, S. Rossignol and S. Grillner, 1988, Neural Control of Rhythmic Motion, \n\n(Wiley, New York). \n\nB.D. Coleman and G.H. Renninger, 1975, J. Theor. BioI. 51, 243. \nB.D. Coleman and G.H. Renninger, 1976, SIAM J. Appl. Math. 31, 111. \nP. R. Conwell, 1987, in Proc. of IEEE First Int. Con! on Neural Networks.III-95. \nH. Gutfreund, J.D. Reg~r and A.P. Young, 1988, J. Phys. A, 21, 2775. \nH. Gutfreund and M. Mezard, 1988, Phys. Rev. Lett. 61, 235. \nK.P. Hadeler and J. Tomiuk, 1977, Arch. Rat. Mech. Anal. 65,87. \nD.O. Hebb, 1949, The Organization of B,ehavior (Wiley, New York). \nJ.1. Hopfield, 1984, Proc. Nat. Acad. Sci. USA 81, 3008. \nD. Kleinfeld, 1984, Proc. Nat. Acad. Sci. USA 83,9469. \nK.E. KUrten and J.W. Clark, 1986, Phys. Lett. 114A, 413. \nM.C. Mackey and L. Glass, 1977, Science 197, 287. \nC.M. Marcus and R.M. Westervelt, 1988, in: Proc. IEEE Con[ on Neural Info. Proc. \n\nSyst .\u2022 Denver. CO. 1987, (American Institute of Physics, New York). \n\nC.M. Marcus and R.M. Westervelt, 1989, Phys. Rev. A 39, 347. \nJ .E. Marsden and M. McCracken, The Hopf Bifurcation and its Applications, (Springer-\n\nVerlag, New York). \n\nU. Riedel, R. KUhn, and J. L. van Hemmen, 1988, Phys. Rev. A 38, 1105. \nS. Shinomoto, 1986, Prog. Theor. Phys. 75, 1313. \nH. Sompolinsky and I. Kanter, 1986, Phys. Rev. Lett. 57, 259. \nH. Sompolinsky, A. erisanti and H.I. Sommers, 1988, Phys. Rev. Lett. 61, 259. \nG. Toulouse, 1977, Commun. Phys. 2, 115. \n\n\f", "award": [], "sourceid": 111, "authors": [{"given_name": "Charles", "family_name": "Marcus", "institution": null}, {"given_name": "R.", "family_name": "Westervelt", "institution": null}]}