Delay induced periodicity in a neural netlet of excitation and inhibition. (English) Zbl 0883.68108

Gopalsamy, K.; Leung, I.

Physica D 89, No. 3-4, 395-426 (1996).

Summary: The dynamical behaviour of a two neuron netlet of excitation and inhibition with a transmission delay is investigated. It is shown that in the absence of delay, the netlet relaxes to the trivial resting state. If the delay is of sufficient magnitude, the network is excited to a temporally periodic cyclic behaviour. The analytical mechanism for the onset of cyclic behaviour is through a Hopf-type bifurcation. Approximate solutions to the periodic output of the netlet is calculated; stability of the temporally periodic cycle is investigated. It is shown that the bifurcation is supercritical. A related discrete version of the continuous time system is formulated. It is found that the discrete system also displays a cyclic behaviour. Results of a number of computer simulations are displayed graphically; the article concludes with a brief neurobiological discussion.

MSC:

68T05	Learning and adaptive systems in artificial intelligence
68U99	Computing methodologies and applications

Keywords:

two neuron netlet; Hopf-type bifurcation

References:

[1]	Amari, S.; Arbib, M. A., Competition and Cooperation in neural networks, (Lecture Notes in Biomathematics, Vol. 45 (1982), Springer: Springer Berlin) · Zbl 0477.00041
[2]	An der Heiden, U., Analysis of neural networks, (Lecture Notes in Biomathematics, Vol. 35 (1980), Springer: Springer Berlin) · Zbl 0591.58021
[3]	Arrowsmith, D. K.; Place, C. M., An introduction to Dynamical systems (1990), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0702.58002
[4]	Babcock, K. L.; Westervelt, R. M., Complex dynamics in simple neural circuits, (Denker, J. S., Neural Networks for Computing (1986), Amer. Inst. Phys: Amer. Inst. Phys New York), 288-293
[5]	Babcock, K. L.; Westervelt, R. M., Dynamics of simple electronic neural networks with added inertia, Physica D, 23, 464-469 (1987)
[6]	Babcock, K. L.; Westervelt, R. M., Dynamics of simple electronic neural networks, Physica D, 28, 305-316 (1986)
[7]	Baird, B., Bifurcation and category learning in network models of oscillating cortex, Physica D, 42, 365-384 (1990)
[8]	Belair, J., Stability in a model of a delayed neural network, J. Dynam. Diff. Eqns., 5, 607-623 (1993) · Zbl 0796.34063
[9]	Belair, J., Stability in delayed neural networks, (Wiener, J.; Hale, J., Ordinary and Delay Differential Equations (1992), Longman Scientific and Technical: Longman Scientific and Technical New York), 6-9 · Zbl 0788.34073
[10]	Bellman, R.; Cooke, K. L., Differential-Difference Equations (1963), Academic Press: Academic Press New York · Zbl 0118.08201
[11]	Bunimovich, L. A.; Sinai, Ya. G., Spacetime chaos in coupled map lattices, Nonlinearity, 1, 491-516 (1988) · Zbl 0679.58028
[12]	Burton, T. A., Neural networks with memory, J. Appl. Math. Stoch. Anal., 4, 313-322 (1991) · Zbl 0746.34043
[13]	Burton, T. A., Averaged neural networks, Neural Networks, 6, 677-680 (1993)
[14]	Chapeau-Blondeau, F.; Chauvet, G., Stable, oscillatory and chaotic regimes in the dynamics of small neural networks with delay, Neural Networks, 5, 735-743 (1992)
[15]	Cohen, M. A., The stability of sustained oscillations in symmetric cooperative competitive networks, Neural Networks, 3, 609-612 (1990)
[16]	Coolen, A. C.C.; Gielen, C. C.A. M., Delays in neural networks, Europhys. Lett., 7, 281-285 (1988)
[17]	Destexhe, A., Stability of periodic oscillations in a network of neurons with time delay, Phys. Lett. A, 187, 309-316 (1994)
[18]	Domany, E.; van Hemmen, J. L.; Schulten, K., Models of Neural Networks (1991), Springer: Springer Berlin · Zbl 0843.92002
[19]	Gopalsamy, K., Stability and Oscillations in Delay Differential Equations of Population Dynamics (1992), Kluwer: Kluwer Dordrecht · Zbl 0752.34039
[20]	Gopalsamy, K.; He, X. Z., Delay independent stability in bidirectional associative memory networks, IEEE Trans. Neural Networks, 5, no. 6 (1994) · Zbl 0807.92020
[21]	Gopalsamy, K.; He, X. Z., Stability in asymmetric Hopfield nets with transmission delays, Physica D, 76, 344-358 (1994) · Zbl 0815.92001
[22]	Gray, C. M.; Konig, P.; Engel, A. K.; Singer, W., (Haken, H.; Stadler, M., Synergetics of Cognition (1990), Springer: Springer New York)
[23]	Halanay, A., Differential Equations, (Mathematics in Science and Engineering, Vol. 23 (1966), Academic Press: Academic Press New York) · Zbl 0695.34054
[24]	Hale, J.; Koçak, H., Dynamics and Bifurcations, (Texts in Applied Mathematics, Vol. 3 (1991), Springer: Springer New York) · Zbl 0745.58002
[25]	Hartline, D. K.; Gassie, D. V., Pattern generation in the lobster (Panulirus) stomatogastric ganglion I. Pyloric neuron kinetics and synaptic interactions, Biol. Cyber, 33, 209-222 (1979)
[26]	Harmon, L. D., Neuromines: Action of a reciprocally inhibitory pair, Science, 146, 1323-1325 (1964)
[27]	Hassard, B. D.; Kazarinoff, N. D.; Wan, Y. H., Theory and Applications of Hopf Bifurcation (1981), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0474.34002
[28]	Hayashi, Y., Oscillatory neural network and learning continuously transformed patterns, Neural Networks, 7, 219-231 (1994)
[29]	Herz, A.; Sulzer, B.; Kühn, R.; van Hemmen, J. L., The Hebb rule: storing static and dynamic objects in an associative network, Europhys. Lett., 7, 663-669 (1988)
[30]	Hopfield, J. J., Neurons with graded response have collective computational properties like those of two state neurons, (Proc. Natl. Acad. Sci., 81 (1984)), 3088-3092 · Zbl 1371.92015
[31]	Kaczmarek, L. K.; Babloyantz, Spatiotemporal patterns in epileptic seizures, Biol. Cybern., 26, 199-208 (1977) · Zbl 0356.92011
[32]	Kleinfield, D.; Sompolinsky, H., Associate neural network model for the generation of temporal patterns, Biophys. J., 54, 1039-1051 (1988)
[33]	Kling, U.; Szekeley, G., Simulation of rhythmic activities I. Function of networks with cyclic inhibitions, Kybernetic, 5, 89-103 (1968) · Zbl 0164.50601
[34]	Llinas, R. R., The intrinsic electrophysiological properties of mammalian neurons: insight into central nervous system function, Science, 242, 1654-1664 (1988)
[35]	Marcus, C. M.; Waugh, F. R.; Westervelt, R. M., Nonlinear dynamics and stability of analog neural networks, Physica D, 51, 234-247 (1991) · Zbl 0800.92059
[36]	Marcus, C. M.; Westervelt, R. M., Dynamics of analog networks with time delay, Advances in Neural Information Processing, (Touretsky, D. S. (1989), Morgan Kauffmann: Morgan Kauffmann San Menlo), 568-576
[37]	Marcus, C. M.; Westervelt, R. M., Stability of analog networks with time delay, Phys. Rev. A, 39, 347-359 (1989)
[38]	Milton, J. G.; Longtin, A.; Beuter, A.; Mackey, M. A.; Glass, L., Complex dynamics and bifurcations in neurology, J. Theor. Biol., 138, 129-147 (1989)
[39]	Muller, B.; Reinhardt, J., Neural Networks (1991), Springer: Springer Berlin
[40]	Nagumo, J.; Sato, S., On the response characteristics of a mathematical neuron model, Kybernetik, 10, 155-164 (1972) · Zbl 0235.92001
[41]	Pavlidis, T., Biological Oscillators (1973), Academic Press: Academic Press London
[42]	Prufer, M., Turbulence in multistep methods for initial value problems, SIAM J. Appl. Math., 45, 32-69 (1985) · Zbl 0576.65067
[43]	Robertson, M.; Pearson, K. G., Neural circuits in the flight system of the locust, J. Neurophysiol., 53, 110-128 (1985)
[44]	Scott, A. C., Neurophysics (1977), Wiley-Interscience: Wiley-Interscience New York
[45]	Sattinter, D. H., Topics in Stability and Bifurcation Theory, (Lecture Notes in Mathematics, Vol. 309 (1973), Springer: Springer Berlin)
[46]	Stech, H. W., The Hopf bifurcation: A stability result and applications, J. Math. Anal. Appl., 71, 525-546 (1979) · Zbl 0418.34073
[47]	Ushiki, S., Central difference scheme and chaos, Physica, 4D, 407-424 (1982) · Zbl 1194.65097
[48]	Vibert, J. F.; Khachayar, P.; Noureddine, A., Interneural delay modification synchronizes biologically plausible neural networks, Neural Networks, 7, 589-607 (1994) · Zbl 0810.92008
[49]	Wiggins, S., Introduction to Applied Nonlinear Dynamical Systems and Chaos, (Texts in Applied Mathematics, Vol. 2 (1990), Springer: Springer New York) · Zbl 1027.37002
[50]	Wilson, D. M.; Waldron, I., Models for the generation of the motor output pattern in flying locusts, (Proc. IEEE, 56 (1968)), 1058-1064
[51]	Wan, Y. H., Computation of the stability condition for the Hopf bifurcation of diffeomorphisms on \(R^2\), SIAM J. Appl. Math., 34, 167-175 (1978) · Zbl 0389.58008
[52]	Wang, X., Period doubling to chaos in a simple neural network: an analytic proof, Complex Systems, 5, 425-441 (1991) · Zbl 0765.58017
[53]	Wang, X.; Bloom, E. K., Discrete-time versus continuous time models of neural networks, J. Comp. Sys. Sci., 45, 1-9 (1992)
[54]	Yamaguti, M.; Matano, H., Euler’s finite difference scheme and chaos, (Proc. Japan Acad. Ser. A, 55 (1979)), 78-80 · Zbl 0434.39003
[55]	Yamaguti, M.; Ushiki, S., Chaos in numerical analysis of ordinary differential equations, Physica, D3, 618-626 (1981) · Zbl 1194.37064
[56]	Yee, H. C.; Sweby, P. K.; Griffiths, D. F., Dynamical approach study of spurious steady-state numerical solutions of nonlinear differential equations I: The dynamics of time discretization and its implications for algorithm development in computational fluid dynamics, J. Comput. Phys., 97, 249-310 (1991) · Zbl 0760.65087