Dynamics of a multiplex neural network with delayed couplings

doi:10.1007/s10483-021-2709-6

Abstract

Abstract: Multiplex networks have drawn much attention since they have been observed in many systems, e.g., brain, transport, and social relationships. In this paper, the nonlinear dynamics of a multiplex network with three neural groups and delayed interactions is studied. The stability and bifurcation of the network equilibrium are discussed, and interesting neural activities of the network are explored. Based on the neuron circuit, transfer function circuit, and time delay circuit, a circuit platform of the network is constructed. It is shown that delayed couplings play crucial roles in the network dynamics, e.g., the enhancement and suppression of the stability, the patterns of the synchronization between networks, and the generation of complicated attractors and multi-stability coexistence.

Key words: neural network, time delay, synchronization, coexisting attractor

2010 MSC Number:

65D15

Xiaochen MAO, Xingyong LI, Weijie DING, Song WANG, Xiangyu ZHOU, Lei QIAO. Dynamics of a multiplex neural network with delayed couplings. Applied Mathematics and Mechanics (English Edition), 2021, 42(3): 441-456.

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References

[1] BATTISTON, F., NICOSIA, V., and LATORA, V. Structural measures for multiplex networks. Physical Review E, 89, 032804(2014)
[2] BOCCALETTI, S., BIANCONI, G., CRIADO, R., DEL GENIO, C. I., GOMEZ-GARDENES, J., ROMANCE, M., SENDINA-NADAL, I., WANG, Z., and ZANIN, M. The structure and dynamics of multilayer networks. Physics Reports, 544, 1-122(2014)
[3] GOMEZ, S., DIAZ-GUILERA, A., GOMEZ-GARDENES, PEREZ-VICENTE, C. J., MORENO, Y., and ARENAS, A. Diffusion dynamics on multiplex networks. Physical Review Letters, 110, 028701(2013)
[4] ZHOU, C. S., ZEMANOVA, L., ZAMORA-LOPEZ, G., HILGETAG, C. C., and KURTHS, J. Structure-function relationship in complex brain networks expressed by hierarchical synchronization. New Journal of Physics, 9, 178(2007)
[5] SUN, W. G., WANG, R. B., WANG, W. X., and CAO, J. T. Analyzing inner and outer synchronization between two coupled discrete-time networks with time delays. Cognitive Neurodynamics, 4, 225-231(2010)
[6] BENTLEY, B., BRANICKY, R., BARNES, C. L., CHEW, Y. L., YEMINI, E., BULLMORE, E. T., VERTES, P. E., and SCHAFER, W. R. The multilayer connectome of Caenorhabditis elegans. PLoS Computational Biology, 12, e1005283(2016)
[7] NIKITIN, D., OMELCHENKO, I., ZAKHAROVA, A., AVETYAN, M., FRADKOV, A. L., and SCHOLL, E. Complex partial synchronization patterns in networks of delay-coupled neurons. Philosophical Transactions of the Royal Society A, 377, 20180128(2019)
[8] CARDILLO, A., ZANIN, M., GOMEZ-GARDENES, ROMANCE, M., GARCIA DEL AMO, A. J., and BOCCALETTI, S. Modeling the multi-layer nature of the European Air Transport Network: resilience and passengers re-scheduling under random failures. European Physical JournalSpecial Topics, 215, 23-33(2013)
[9] BATTISTON, F., NICOSIA, V., CHAVEZ, M., and LATORA, V. Multilayer motif analysis of brain networks. Chaos, 27, 047404(2017)
[10] KORN, H. and FAURE, P. Is there chaos in the brain? II. experimental evidence and related models. Comptes Rendus Biologies, 326, 787-840(2003)
[11] HU, X. Y., LIU, C. X., LIU, L., NI, J. K., and YAO, Y. P. Chaotic dynamics in a neural network under electromagnetic radiation. Nonlinear Dynamics, 91, 1541-1554(2018)
[12] ZHOU, L. L., TAN, F., YU, F., and LIU, W. Cluster synchronization of two-layer nonlinearly coupled multiplex networks with multi-links and time-delays. Neurocomputing, 359, 264-275(2019)
[13] MAJHI, S., PERC, M., and GHOSH, D. Chimera states in a multilayer network of coupled and uncoupled neurons. Chaos, 27, 073109(2017)
[14] YU, Y. and WANG, Q. Y. Oscillation dynamics in an extended model of thalamic-basal ganglia. Nonlinear Dynamics, 98, 1065-1080(2019)
[15] MA, J., YANG, Z. Q., YANG, L. J., and TANG, J. A physical view of computational neurodynamics. Journal of Zhejiang University-Science A, 20, 639-659(2019)
[16] ZHOU, J., LIU, Z. R., and XIANG, L. Global dynamics of delayed bidirectional associative memory (BAM) neural networks. Applied Mathematics and Mechanics (English Edition), 26(3), 327-335(2005) https://doi.org/10.1007/BF02440083
[17] KACHHVAH, A. D. and JALAN, S. Delay regulated explosive synchronization in multiplex networks. New Journal of Physics, 21, 015006(2019)
[18] SAWICKI, J., OMELCHENKO, I., ZAKHAROVA, A., and SCHOELL, E. Delay controls chimera relay synchronization in multiplex networks. Physical Review E, 98, 062224(2018)
[19] HAMMOND, C., BERGMAN, H., and BROWN, P. Pathological synchronization in Parkinson’s disease: networks, models and treatments. Trends in Neurosciences, 30, 357-364(2007)
[20] MAO, X. C. and WANG, Z. H. Stability, bifurcation, and synchronization of delay-coupled ring neural networks. Nonlinear Dynamics, 84, 1063-1078(2016)
[21] TANG, Y., QIAN, F., GAO, H. J., and KURTHS, J. Synchronization in complex networks and its application — a survey of recent advances and challenges. Annual Reviews in Control, 38, 184-198(2014)
[22] HAN, F., GU, X. C., WANG, Z. J., FAN, H., CAO, J. F., and LU, Q. S. Global firing rate contrast enhancement in E/I neuronal networks by recurrent synchronized inhibition. Chaos, 28, 106324(2018)
[23] FAN, D. G., ZHENG, Y. H., YANG, Z. C., and WANG, Q. Y. Improving control effects of absence seizures using single-pulse alternately resetting stimulation (SARS) of corticothalamic circuit. Applied Mathematics and Mechanics (English Edition), 41(9), 1287-1302(2020) https://doi.org/10.1007/s10483-020-2644-8
[24] LIANG, S. and WANG, Z. H. Controlling a neuron by stimulating a coupled neuron. Applied Mathematics and Mechanics (English Edition), 40(1), 13-24(2019) https://doi.org/10.1007/s10483- 019-2407-8
[25] HU, H. Y. and WANG, Z. H. Dynamics of Controlled Mechanical Systems with Delayed Feedback, Springer-Verlag, Heidelberg (2002)
[26] CHEN, L. X. and CAI, G. P. Design method of multiple time-delay controller for active structural vibration control. Applied Mathematics and Mechanics (English Edition), 30(11), 1405-1414(2009) https://doi.org/10.1007/s10483-009-1106-z
[27] STEPAN, G. Delay effects in brain dynamics. Philosophical Transactions of the Royal Society A, 367, 1059-1062(2009)
[28] FLUNKERT, V., FISCHER, I., and SCHOELL, E. Dynamics, control and information in delaycoupled systems. Philosophical Transactions of the Royal Society A, 371, 20120465(2013)
[29] WANG, Y. P., CONG, Y. H., and HU, G. D. Delay-dependent stability of linear multistep methods for differential systems with distributed delays. Applied Mathematics and Mechanics (English Edition), 39(12), 1837-1844(2018) https://doi.org/10.1007/s10483-018-2392-9
[30] BALDI, P. and ATIYA, A. F. How delays affect neural dynamics and learning. IEEE Transactions on Neural Networks, 5, 612-621(1994)
[31] MAO, X. C., SUN, J. Q., and LI, S. F. Dynamics of delay-coupled FitzHugh-Nagumo neural rings. Chaos, 28, 013104(2018)
[32] CAMPBELL, S. A., EDWARDS, R., and VAN DEN DRIESSCHE, P. Delayed coupling between two neural network loops. SIAM Journal on Applied Mathematics, 65, 316-335(2005)
[33] HSU, C. H. and YANG, T. S. Periodic oscillations arising and death in delay-coupled neural loops. International Journal of Bifurcation and Chaos, 17, 4015-4032(2007)
[34] MAO, X. C., ZHOU, X. Y., SHI, T. T., and QIAO, L. Dynamical analysis of coupled bidirectional FitzHugh-Nagumo neuronal networks with multiple delays. Journal of Computational and Nonlinear Dynamics, 14, 061002(2019)
[35] SINGH, A., GHOSH, S., JALAN, S., and KURTHS, J. Synchronization in delayed multiplex networks. Europhysics Letters, 111, 30010(2015)
[36] CHENG, C. Y. Induction of Hopf bifurcation and oscillation death by delays in coupled networks. Physics Letters A, 374, 178-185(2009)
[37] SONG, Y. L. and XU, J. Inphase and antiphase synchronization in a delay-coupled system with applications to a delay-coupled FitzHugh-Nagumo system. IEEE Transactions on Neural Networks and Learning Systems, 23, 1659-1670(2012)
[38] XU, X., YU, D., and WANG, Z. Inter-layer synchronization of periodic solutions in two coupled rings with time delay. Physica D, 396, 1-11(2019)
[39] HOPFIELD, J. J. Neurons with graded response have collective computational properties like those of two-state neurons. Proceedings of the National Academy of Sciences of the United States of American, 81, 3088-3092(1984)
[40] HASSARD, B. D., KAZARINOFF, N. D., and WAN, Y. H. Theory and Application of Hopf Bifurcation, Cambridge University Press, Cambridge (1981)
[41] DUAN, S. K. and LIAO, X. F. An electronic implementation for Liao’s chaotic delayed neuron model with non-monotonous activation function. Physics Letters A, 369, 37-43(2007)
[42] BAO, B. C., QIAN, H., XU, Q., CHEN, M., WANG, J., and YU, Y. J. Coexisting behaviors of asymmetric attractors in hyperbolic-type memristor based Hopfield neural network. Frontiers in Computational Neuroscience, 11, 81(2017)
[43] ABLAY, G. Novel chaotic delay systems and electronic circuit solutions. Nonlinear Dynamics, 81, 1795-1804(2015)