Skip to main content
SpringerLink
Log in

Neurodynamics

  • Chapter
Springer Handbook of Computational Intelligence

Part of the book series: Springer Handbooks ((SHB))

Abstract

This chapter introduces basic concepts, phenomena, and properties of neurodynamic systems. it consists of four sections with the first two on various neurodynamic behaviors of general neurodynamics and the last two on two types of specific neurodynamic systems. The neurodynamic behaviors discussed in the first two sections include attractivity, oscillation, synchronization, and chaos. The two specific neurodynamics systems are memrisitve neurodynamic systems and neurodynamic optimization

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 269.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 349.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Abbreviations

AM:

amplitude modulation

ANN:

artificial neural network

APSD:

auto power spectral density

ASIC:

application-specific integrated circuit

CCF:

cross correlation function

CG:

Cohen–Grossberg

CML:

coupled map lattice

CMOS:

complementary metal-oxide-semiconductor

CNN:

cellular neural network

CPSD:

cross power spectral density

CPU:

central processing unit

EEG:

electroencephalogram

HH:

Hodgkin–Huxley

LFP:

local field potential

LMI:

linear matrix inequalities

MEG:

magnetoencephalogram

MNN:

memristor-based neural network

MRNN:

memristor-based recurrent neural network

NDS:

nonlinear dynamical systems

ODE:

ordinary differential equation

PDE:

partial differential equation

PLV:

phase lock value

PV:

principal value

SOC:

self-organized criticality

SR:

stochastic resonance

STDP:

spike-timing dependent learning

TTGA:

trainable threshold gate array

VLSI:

very large scale integration

WC:

Wilson–Cowan

References

  1. R. Abraham: Dynamics: The Geometry of Behavior (Aerial, Santa Cruz 1982)

    Google Scholar 

  2. J. Robinson: Attractor. In: Encyclopedia of Nonlinear Science, ed. by A. Scott (Routledge, New York 2005) pp. 26–28

    Google Scholar 

  3. S. Grossberg: Nonlinear difference-differential equations in prediction and learning theory, Proc. Natl. Acad. Sci. 58, 1329–1334 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  4. S. Grossberg: Global ratio limit theorems for some nonlinear functional differential equations I, Bull. Am. Math. Soc. 74, 93–100 (1968)

    MathSciNet  MATH  Google Scholar 

  5. H. Zhang, Z. Wang, D. Liu: Robust exponential stability of recurrent neural networks with multiple time-varying delays, IEEE Trans. Circuits Syst. II: Express Br. 54, 730–734 (2007)

    Article  MathSciNet  Google Scholar 

  6. A.N. Michel, K. Wang, D. Liu, H. Ye: Qualitative limitations incurred in implementations of recurrent neural networks, IEEE Cont. Syst. Mag. 15(3), 52–65 (1995)

    Article  Google Scholar 

  7. H. Zhang, Z. Wang, D. Liu: Global asymptotic stability of recurrent neural networks with multiple time varying delays, IEEE Trans. Neural Netw. 19(5), 855–873 (2008)

    Article  Google Scholar 

  8. S. Hu, D. Liu: On the global output convergence of a class of recurrent neural networks with time-varying inputs, Neural Netw. 18(2), 171–178 (2005)

    Article  MATH  Google Scholar 

  9. D. Liu, S. Hu, J. Wang: Global output convergence of a class of continuous-time recurrent neural networks with time-varying thresholds, IEEE Trans. Circuits Syst. II: Express Br. 51(4), 161–167 (2004)

    Article  Google Scholar 

  10. H. Zhang, Z. Wang, D. Liu: Robust stability analysis for interval Cohen–Grossberg neural networks with unknown time varying delays, IEEE Trans. Neural Netw. 19(11), 1942–1955 (2008)

    Article  Google Scholar 

  11. M. Han, J. Fan, J. Wang: A dynamic feedforward neural network based on Gaussian particle swarm optimization and its application for predictive control, IEEE Trans. Neural Netw. 22(9), 1457–1468 (2011)

    Article  Google Scholar 

  12. S. Mehraeen, S. Jagannathan, M.L. Crow: Decentralized dynamic surface control of large-scale interconnected systems in strict-feedback form using neural networks with asymptotic stabilization, IEEE Trans. Neural Netw. 22(11), 1709–1722 (2011)

    Article  Google Scholar 

  13. Y. Zhang, T. Chai, H. Wang: A nonlinear control method based on anfis and multiple models for a class of SISO nonlinear systems and its application, IEEE Trans. Neural Netw. 22(11), 1783–1795 (2011)

    Article  Google Scholar 

  14. Y. Chen, W.X. Zheng: Stability and L 2 performance analysis of stochastic delayed neural networks, IEEE Trans. Neural Netw. 22(10), 1662–1668 (2011)

    Article  Google Scholar 

  15. M. Di Marco, M. Grazzini, L. Pancioni: Global robust stability criteria for interval delayed full-range cellular neural networks, IEEE Trans. Neural Netw. 22(4), 666–671 (2011)

    Article  Google Scholar 

  16. W.-H. Chen, W.X. Zheng: A new method for complete stability analysis of cellular neural networks with time delay, IEEE Trans. Neural Netw. 21(7), 1126–1139 (2010)

    Article  Google Scholar 

  17. H. Zhang, Z. Wang, D. Liu: Global asymptotic stability and robust stability of a general class of Cohen–Grossberg neural networks with mixed delays, IEEE Trans. Circuits Syst. I: Regul. Pap. 56(3), 616–629 (2009)

    Article  MathSciNet  Google Scholar 

  18. X.X. Liao, J. Wang: Algebraic criteria for global exponential stability of cellular neural networks with multiple time delays, IEEE Trans. Circuits Syst. I 50, 268–275 (2003)

    Article  MathSciNet  Google Scholar 

  19. Z.G. Zeng, J. Wang, X.X. Liao: Global exponential stability of a general class of recurrent neural networks with time-varying delays, IEEE Trans. Circuits Sys. I 50(10), 1353–1358 (2003)

    Article  MathSciNet  Google Scholar 

  20. D. Angeli: Multistability in systems with counter-clockwise input-output dynamics, IEEE Trans. Autom. Control 52(4), 596–609 (2007)

    Article  MathSciNet  Google Scholar 

  21. D. Angeli: Systems with counterclockwise input-output dynamics, IEEE Trans. Autom. Control 51(7), 1130–1143 (2006)

    Article  MathSciNet  Google Scholar 

  22. D. Angeli: Convergence in networks with counterclockwise neural dynamics, IEEE Trans. Neural Netw. 20(5), 794–804 (2009)

    Article  Google Scholar 

  23. J. Saez-Rodriguez, A. Hammerle-Fickinger, O. Dalal, S. Klamt, E.D. Gilles, C. Conradi: Multistability of signal transduction motifs, IET Syst. Biol. 2(2), 80–93 (2008)

    Article  Google Scholar 

  24. L. Chandrasekaran, V. Matveev, A. Bose: Multistability of clustered states in a globally inhibitory network, Phys. D 238(3), 253–263 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  25. B.K. Goswami: Control of multistate hopping intermittency, Phys. Rev. E 78(6), 066208 (2008)

    Article  MathSciNet  Google Scholar 

  26. A. Rahman, M.K. Sanyal: The tunable bistable and multistable memory effect in polymer nanowires, Nanotechnology 19(39), 395203 (2008)

    Article  Google Scholar 

  27. K.C. Tan, H.J. Tang, W.N. Zhang: Qualitative analysis for recurrent neural networks with linear threshold transfer functions, IEEE Trans. Circuits Syst. I: Regul. Pap. 52(5), 1003–1012 (2005)

    Article  MathSciNet  Google Scholar 

  28. H.J. Tang, K.C. Tan, E.J. Teoh: Dynamics analysis and analog associative memory of networks with LT neurons, IEEE Trans. Neural Netw. 17(2), 409–418 (2006)

    Article  Google Scholar 

  29. L. Zou, H.J. Tang, K.C. Tan, W.N. Zhang: Nontrivial global attractors in 2-D multistable attractor neural networks, IEEE Trans. Neural Netw. 20(11), 1842–1851 (2009)

    Article  Google Scholar 

  30. D. Liu, A.N. Michel: Sparsely interconnected neural networks for associative memories with applications to cellular neural networks, IEEE Trans. Circuits Syst. II: Analog Digit, Signal Process. 41(4), 295–307 (1994)

    MathSciNet  Google Scholar 

  31. M. Brucoli, L. Carnimeo, G. Grassi: Discrete-time cellular neural networks for associative memories with learning and forgetting capabilities, IEEE Trans. Circuits Syst. I: Fundam. Theory Appl. 42(7), 396–399 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  32. R. Perfetti: Dual-mode space-varying self-designing cellular neural networks for associative memory, IEEE Trans. Circuits Syst. I: Fundam. Theory Appl. 46(10), 1281–1285 (1999)

    Article  Google Scholar 

  33. G. Grassi: On discrete-time cellular neural networks for associative memories, IEEE Trans. Circuits Syst. I: Fundam. Theory Appl. 48(1), 107–111 (2001)

    Article  MATH  Google Scholar 

  34. L. Wang, X. Zou: Capacity of stable periodic solutions in discrete-time bidirectional associative memory neural networks, IEEE Trans. Circuits Syst. II: Express Br. 51(6), 315–319 (2004)

    Article  Google Scholar 

  35. J. Milton: Epilepsy: Multistability in a dynamic disease. In: Self- Organized Biological Dynamics Nonlinear Control: Toward Understanding Complexity, Chaos, and Emergent Function in Living Systems, ed. by J. Walleczek (Cambridge Univ. Press, Cambridge 2000) pp. 374–386

    Chapter  Google Scholar 

  36. U. Feudel: Complex dynamics in multistable systems, Int. J. Bifurc. Chaos 18(6), 1607–1626 (2008)

    Article  MathSciNet  Google Scholar 

  37. J. Hizanidis, R. Aust, E. Scholl: Delay-induced multistability near a global bifurcation, Int. J. Bifurc. Chaos 18(6), 1759–1765 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  38. G.G. Wells, C.V. Brown: Multistable liquid crystal waveplate, Appl. Phys. Lett. 91(22), 223506 (2007)

    Article  Google Scholar 

  39. G. Deco, D. Marti: Deterministic analysis of stochastic bifurcations in multi-stable neurodynamical systems, Biol. Cybern. 96(5), 487–496 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  40. J.D. Cao, G. Feng, Y.Y. Wang: Multistability and multiperiodicity of delayed Cohen-Grossberg neural networks with a general class of activation functions, Phys. D 237(13), 1734–1749 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  41. C.Y. Cheng, K.H. Lin, C.W. Shih: Multistability in recurrent neural networks, SIAM J. Appl. Math. 66(4), 1301–1320 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  42. Z. Yi, K.K. Tan: Multistability of discrete-time recurrent neural networks with unsaturating piecewise linear activation functions, IEEE Trans. Neural Netw. 15(2), 329–336 (2004)

    Article  Google Scholar 

  43. Z. Yi, K.K. Tan, T.H. Lee: Multistability analysis for recurrent neural networks with unsaturating piecewise linear transfer functions, Neural Comput. 15(3), 639–662 (2003)

    Article  MATH  Google Scholar 

  44. Z.G. Zeng, T.W. Huang, W.X. Zheng: Multistability of recurrent neural networks with time-varying delays and the piecewise linear activation function, IEEE Trans. Neural Netw. 21(8), 1371–1377 (2010)

    Article  Google Scholar 

  45. Z.G. Zeng, J. Wang, X.X. Liao: Stability analysis of delayed cellular neural networks described using cloning templates, IEEE Trans. Circuits Syst. I: Regul. Pap. 51(11), 2313–2324 (2004)

    Article  MathSciNet  Google Scholar 

  46. Z.G. Zeng, J. Wang: Multiperiodicity and exponential attractivity evoked by periodic external inputs in delayed cellular neural networks, Neural Comput. 18(4), 848–870 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  47. L.L. Wang, W.L. Lu, T.P. Chen: Multistability and new attraction basins of almost-periodic solutions of delayed neural networks, IEEE Trans. Neural Netw. 20(10), 1581–1593 (2009)

    Article  Google Scholar 

  48. G. Huang, J.D. Cao: Delay-dependent multistability in recurrent neural networks, Neural Netw. 23(2), 201–209 (2010)

    Article  Google Scholar 

  49. L.L. Wang, W.L. Lu, T.P. Chen: Coexistence and local stability of multiple equilibria in neural networks with piecewise linear nondecreasing activation functions, Neural Netw. 23(2), 189–200 (2010)

    Article  Google Scholar 

  50. L. Zhang, Z. Yi, J.L. Yu, P.A. Heng: Some multistability properties of bidirectional associative memory recurrent neural networks with unsaturating piecewise linear transfer functions, Neurocomputing 72(16–18), 3809–3817 (2009)

    Article  Google Scholar 

  51. X.B. Nie, J.D. Cao: Multistability of competitive neural networks with time-varying and distributed delays, Nonlinear Anal.: Real World Appl. 10(2), 928–942 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  52. C.Y. Cheng, K.H. Lin, C.W. Shih: Multistability and convergence in delayed neural networks, Phys. D 225(1), 61–74 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  53. T.J. Sejnowski, C. Koch, P.S. Churchland: Computational neuroscience, Science 241(4871), 1299 (1988)

    Article  Google Scholar 

  54. G. Edelman: Remembered Present: A Biological Theory of Consciousness (Basic Books, New York 1989)

    Google Scholar 

  55. W.J. Freeman: Societies of Brains: A Study in the Neuroscience of Love and Hate (Lawrence Erlbaum, New York 1995)

    Google Scholar 

  56. R. Llinas, U. Ribary, D. Contreras, C. Pedroarena: The neuronal basis for consciousness, Philos. Trans. R. Soc. B 353(1377), 1841 (1998)

    Article  Google Scholar 

  57. F. Crick, C. Koch: A framework for consciousness, Nat. Neurosci. 6(2), 119–126 (2003)

    Article  Google Scholar 

  58. A.L. Hodgkin, A.F. Huxley: A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol. 117(4), 500 (1952)

    Article  Google Scholar 

  59. A. Pikovsky, M. Rosenblum: Synchronization, Scholarpedia 2(12), 1459 (2007)

    Article  MATH  Google Scholar 

  60. D. Golomb, A. Shedmi, R. Curtu, G.B. Ermentrout: Persistent synchronized bursting activity in cortical tissues with low magnesium concentration: A modeling study, J. Neurophysiol. 95(2), 1049–1067 (2006)

    Article  Google Scholar 

  61. M.L.V. Quyen, J. Foucher, J.-P. Lachaux, E. Rodriguez, A. Lutz, J. Martinerie, F.J. Varela: Comparison of Hilbert transform and wavelet methods for the analysis of neuronal synchrony, J. Neurosci. Methods 111(2), 83–98 (2001)

    Article  Google Scholar 

  62. W.J. Freeman, L.J. Rogers: Fine temporal resolution of analytic phase reveals episodic synchronization by state transitions in gamma EEGs, J. Neurophysiol. 87(2), 937–945 (2002)

    Google Scholar 

  63. G.E.P. Box, G.M. Jenkins, G.C. Reinsel: Ser. Probab. Stat, Time Series Analysis: Forecasting and Control, Vol. 734 (Wiley, Hoboken 2008)

    Google Scholar 

  64. R.W. Thatcher, D.M. North, C.J. Biver: Development of cortical connections as measured by EEG coherence and phase delays, Hum. Brain Mapp. 29(12), 1400–1415 (2007)

    Article  Google Scholar 

  65. A. Pikovsky, M. Rosenblum, J. Kurths: Synchronization: A Universal Concept in Nonlinear Sciences, Vol. 12 (Cambridge Univ. Press, Cambridge 2003)

    MATH  Google Scholar 

  66. J. Rodriguez, R. Kozma: Phase synchronization in mesoscopic electroencephalogram arrays. In: Intelligent Engineering Systems Through Artificial Neural Networks Series, ed. by C. Dagli (ASME, New York 2007) pp. 9–14

    Chapter  Google Scholar 

  67. J.M. Barrie, W.J. Freeman, M.D. Lenhart: Spatiotemporal analysis of prepyriform, visual, auditory, and somesthetic surface EEGs in trained rabbits, J. Neurophysiol. 76(1), 520–539 (1996)

    Google Scholar 

  68. G. Dumas, M. Chavez, J. Nadel, J. Martinerie: Anatomical connectivity influences both intra-and inter-brain synchronizations, PloS ONE 7(5), e36414 (2012)

    Google Scholar 

  69. J.A.S. Kelso: Dynamic Patterns: The Self-Organization of Brain and Behavior (MIT Press, Cambridge 1995)

    Google Scholar 

  70. S. Campbell, D. Wang: Synchronization and desynchronization in a network of locally coupled Wilson–Cowan oscillators, IEEE Trans. Neural Netw. 7(3), 541–554 (1996)

    Article  Google Scholar 

  71. H. Kurokawa, C.Y. Ho: A learning rule of the oscillatory neural networks for in-phase oscillation, IEICE Trans. Fundam. Electron. Commun. Comput. Sci. 80(9), 1585–1594 (1997)

    Google Scholar 

  72. G. Buzsaki: Rhythms of the Brain (Oxford Univ. Press, New York 2009)

    MATH  Google Scholar 

  73. A.K. Engel, P. Fries, W. Singer: Dynamic predictions: Oscillations and synchrony in top-down processing, Nat. Rev. Neurosci. 2(10), 704–716 (2001)

    Article  Google Scholar 

  74. W.J. Freeman, R.Q. Quiroga: Imaging Brain Function with EEG: Advanced Temporal and Spatial Analysis of Electroencephalographic Signals (Springer, New York 2013)

    Book  Google Scholar 

  75. H. Haken: Cooperative phenomena in systems far from thermal equilibrium and in nonphysical systems, Rev. Mod. Phys. 47(1), 67 (1975)

    Article  MathSciNet  Google Scholar 

  76. S.H. Strogatz: Exploring complex networks, Nature 410(6825), 268–276 (2001)

    Article  Google Scholar 

  77. O. Sporns, D.R. Chialvo, M. Kaiser, C.C. Hilgetag: Organization, development and function of complex brain networks, Trends Cogn. Sci. 8(9), 418–425 (2004)

    Article  Google Scholar 

  78. B. Bollobás, R. Kozma, D. Miklos (Eds.): Handbook of Large-Scale Random Networks, Bolyai Soc. Math. Stud., Vol. 18 (Springer, Berlin, Heidelberg 2009)

    MATH  Google Scholar 

  79. Y. Kuramoto: Cooperative dynamics of oscillator community, Prog. Theor. Phys. Suppl. 79, 223–240 (1984)

    Article  Google Scholar 

  80. M.G. Rosenblum, A.S. Pikovsky: Controlling synchronization in an ensemble of globally coupled oscillators, Phys. Rev. Lett. 92(11), 114102 (2004)

    Article  Google Scholar 

  81. O.V. Popovych, P.A. Tass: Synchronization control of interacting oscillatory ensembles by mixed nonlinear delayed feedback, Phys. Rev. E 82(2), 026204 (2010)

    Article  MathSciNet  Google Scholar 

  82. W.J. Freeman: The physiology of perception, Sci. Am. 264, 78–85 (1991)

    Article  Google Scholar 

  83. M.A. Cohen, S. Grossberg: Absolute stability of global pattern formation and parallel memory storage by competitive neural networks, IEEE Trans. Syst. Man Cybern. 13(5), 815–826 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  84. J.J. Hopfield, D.W. Tank: Computing with neural circuits – A model, Science 233(4764), 625–633 (1986)

    Article  Google Scholar 

  85. C.M. Marcus, R.M. Westervelt: Dynamics of iterated-map neural networks, Phys. Rev. A 40(1), 501 (1989)

    Article  MathSciNet  Google Scholar 

  86. W. Yu, J. Cao, J. Wang: An LMI approach to global asymptotic stability of the delayed Cohen-Grossberg neural network via nonsmooth analysis, Neural Netw. 20(7), 810–818 (2007)

    Article  MATH  Google Scholar 

  87. F.C. Hoppensteadt, E.M. Izhikevich: Weakly Connected Neural Networks, Applied Mathematical Sciences, Vol. 126 (Springer, New York 1997)

    MATH  Google Scholar 

  88. H.R. Wilson, J.D. Cowan: Excitatory and inhibitory interactions in localized populations of model neurons, Biophys. J. 12(1), 1–24 (1972)

    Article  Google Scholar 

  89. H.R. Wilson, J.D. Cowan: A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue, Biol. Cybern. 13(2), 55–80 (1973)

    MATH  Google Scholar 

  90. P.C. Bressloff: Spatiotemporal dynamics of continuum neural fields, J. Phys. A: Math. Theor. 45(3), 033001 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  91. D. Wang: Object selection based on oscillatory correlation, Neural Netw. 12(4), 579–592 (1999)

    Article  Google Scholar 

  92. A. Renart, R. Moreno-Bote, X.-J. Wang, N. Parga: Mean-driven and fluctuation-driven persistent activity in recurrent networks, Neural Comput. 19(1), 1–46 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  93. M. Ursino, E. Magosso, C. Cuppini: Recognition of abstract objects via neural oscillators: interaction among topological organization, associative memory and gamma band synchronization, IEEE Trans. Neural Netw. 20(2), 316–335 (2009)

    Article  Google Scholar 

  94. W.J. Freeman: Mass Action in the Nervous System (Academic, New York 1975)

    Google Scholar 

  95. D. Xu, J. Principe: Dynamical analysis of neural oscillators in an olfactory cortex model, IEEE Trans. Neural Netw. 15(5), 1053–1062 (2004)

    Article  Google Scholar 

  96. R. Ilin, R. Kozma: Stability of coupled excitatory–inhibitory neural populations and application to control of multi-stable systems, Phys. Lett. A 360(1), 66–83 (2006)

    Article  Google Scholar 

  97. R. Ilin, R. Kozma: Control of multi-stable chaotic neural networks using input constraints, 2007. IJCNN 2007. Int. Jt. Conf. Neural Netw., Orlando (2007) pp. 2194–2199

    Google Scholar 

  98. G. Deco, V. K. Jirsa, P. A. Robinson, M. Breakspear, K. Friston: The dynamic brain: From spiking neurons to neural masses and cortical fields, PLoS Comput. Biol. 4(8), e1000092 (2008)

    Google Scholar 

  99. L. Ingber: Generic mesoscopic neural networks based on statistical mechanics of neocortical interactions, Phys. Rev. A 45(4), 2183–2186 (1992)

    Article  Google Scholar 

  100. V.K. Jirsa, K.J. Jantzen, A. Fuchs, J.A. Scott Kelso: Spatiotemporal forward solution of the EEG and meg using network modeling, IEEE Trans. Med. Imaging 21(5), 493–504 (2002)

    Article  Google Scholar 

  101. S. Coombes, C. Laing: Delays in activity-based neural networks, Philos. Trans. R. Soc. A 367(1891), 1117–1129 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  102. V.K. Jirsa: Neural field dynamics with local and global connectivity and time delay, Philos. Trans. R. Soc. A 367(1891), 1131–1143 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  103. K. Kaneko: Clustering, coding, switching, hierarchical ordering, and control in a network of chaotic elements, Phys. D 41(2), 137–172 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  104. R. Kozma: Intermediate-range coupling generates low-dimensional attractors deeply in the chaotic region of one-dimensional lattices, Phys. Lett. A 244(1), 85–91 (1998)

    Article  Google Scholar 

  105. S. Ishii, M.-A. Sato: Associative memory based on parametrically coupled chaotic elements, Phys. D 121(3), 344–366 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  106. F. Moss, A. Bulsara, M.F. Schlesinger (Eds.): The proceedings of the NATO Advanced Research Workshop: Stochastic Resonance in Physics and Biology (Plenum Press, New York 1993)

    Google Scholar 

  107. S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes: Critical phenomena in complex networks, Rev. Mod. Phys. 80(4), 1275 (2008)

    Article  Google Scholar 

  108. M.D. McDonnell, L.M. Ward: The benefits of noise in neural systems: bridging theory and experiment, Nat. Rev. Neurosci. 12(7), 415–426 (2011)

    Article  Google Scholar 

  109. A.V. Goltsev, M.A. Lopes, K.-E. Lee, J.F.F. Mendes: Critical and resonance phenomena in neural networks, arXiv preprint arXiv:1211.5686 (2012)

    Google Scholar 

  110. P. Bak: How Nature Works: The Science of Self-Organized Criticality (Copernicus, New York 1996)

    Book  MATH  Google Scholar 

  111. J.M. Beggs, D. Plenz: Neuronal avalanches in neocortical circuits, J. Neurosci. 23(35), 11167–11177 (2003)

    Google Scholar 

  112. J.M. Beggs: The criticality hypothesis: How local cortical networks might optimize information processing, Philos. Trans. R. Soc. A 366(1864), 329–343 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  113. T. Petermann, T.C. Thiagarajan, M.A. Lebedev, M.A.L. Nicolelis, D.R. Chialvo, D. Plenz: Spontaneous cortical activity in awake monkeys composed of neuronal avalanches, Proc. Natl. Acad. Sci. 106(37), 15921–15926 (2009)

    Article  Google Scholar 

  114. M. Puljic, R. Kozma: Narrow-band oscillations in probabilistic cellular automata, Phys. Rev. E 78(2), 026214 (2008)

    Article  Google Scholar 

  115. R. Kozma, M. Puljic, W.J. Freeman: Thermodynamic model of criticality in the cortex based on EEG/ECoG data. In: Criticality in Neural Systems, ed. by D. Plenz, E. Niebur (Wiley, Hoboken 2014) pp. 153–176

    Chapter  Google Scholar 

  116. J.-P. Eckmann, D. Ruelle: Ergodic theory of chaos and strange attractors, Rev. Mod. Phys. 57(3), 617 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  117. E. Ott, C. Grebogi, J.A. Yorke: Controlling chaos, Phys. Rev. Lett. 64(11), 1196–1199 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  118. E.N. Lorenz: Deterministic nonperiodic flow, J. Atmos. Sci. 20(2), 130–141 (1963)

    Article  Google Scholar 

  119. K. Aihara, T. Takabe, M. Toyoda: Chaotic neural networks, Phys. Lett. A 144(6), 333–340 (1990)

    Article  MathSciNet  Google Scholar 

  120. K. Aihara, H. Suzuki: Theory of hybrid dynamical systems and its applications to biological and medical systems, Philos. Trans. R. Soc. A 368(1930), 4893–4914 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  121. G. Matsumoto, K. Aihara, Y. Hanyu, N. Takahashi, S. Yoshizawa, J.-I. Nagumo: Chaos and phase locking in normal squid axons, Phys. Lett. A 123(4), 162–166 (1987)

    Article  Google Scholar 

  122. K. Aihara: Chaos engineering and its application to parallel distributed processing with chaotic neural networks, Proc. IEEE 90(5), 919–930 (2002)

    Article  MathSciNet  Google Scholar 

  123. L. Wang, S. Li, F. Tian, X. Fu: A noisy chaotic neural network for solving combinatorial optimization problems: Stochastic chaotic simulated annealing, IEEE Trans. Syst. Man Cybern. B 34(5), 2119–2125 (2004)

    Article  Google Scholar 

  124. Z. Zeng, J. Wang: Improved conditions for global exponential stability of recurrent neural networks with time-varying delays, IEEE Trans. Neural Netw. 17(3), 623–635 (2006)

    Article  MathSciNet  Google Scholar 

  125. M.D. Marco, M. Grazzini, L. Pancioni: Global robust stability criteria for interval delayed full-range cellular neural networks, IEEE Trans. Neural Netw. 22(4), 666–671 (2011)

    Article  Google Scholar 

  126. C.A. Skarda, W.J. Freeman: How brains make chaos in order to make sense of the world, Behav. Brain Sci. 10(2), 161–195 (1987)

    Article  Google Scholar 

  127. H.D.I. Abarbanel, M.I. Rabinovich, A. Selverston, M.V. Bazhenov, R. Huerta, M.M. Sushchik, L.L. Rubchinskii: Synchronisation in neural networks, Phys.-Usp. 39(4), 337–362 (1996)

    Article  Google Scholar 

  128. H. Korn, P. Faure: Is there chaos in the brain? II. experimental evidence and related models, c.r. Biol. 326(9), 787–840 (2003)

    Article  Google Scholar 

  129. R. Kozma, W.J. Freeman: Intermittent spatio-temporal desynchronization and sequenced synchrony in ECoG signals, Chaos Interdiscip. J. Nonlinear Sci. 18(3), 037131 (2008)

    Article  Google Scholar 

  130. K. Kaneko: Collapse of Tori and Genesis of Chaos in Dissipative Systems (World Scientific Publ., Singapore 1986)

    Book  MATH  Google Scholar 

  131. K. Aihara: Chaos in neural networks. In: The Impact of Chaos on Science and Society, ed. by C. Grebogi, J.A. Yorke (United Nations Publ., New York 1997) pp. 110–126

    Google Scholar 

  132. I. Tsuda: Toward an interpretation of dynamic neural activity in terms of chaotic dynamical systems, Behav. Brain Sci. 24(5), 793–809 (2001)

    Article  Google Scholar 

  133. H. Bersini, P. Sener: The connections between the frustrated chaos and the intermittency chaos in small Hopfield networks, Neural Netw. 15(10), 1197–1204 (2002)

    Article  Google Scholar 

  134. P. Berge, Y. Pomeau, C. Vidal: Order in Chaos (Herman, Paris and Wiley, New York 1984)

    MATH  Google Scholar 

  135. Y. Pomeau, P. Manneville: Intermittent transition to turbulence in dissipative dynamical systems, Commun. Math. Phys. 74(2), 189–197 (1980)

    Article  MathSciNet  Google Scholar 

  136. T. Higuchi: Relationship between the fractal dimension and the power law index for a time series: A numerical investigation, Phys. D 46(2), 254–264 (1990)

    Article  MATH  Google Scholar 

  137. B. Mandelbrot: Fractals and Chaos: The Mandelbrot Set and Beyond, Vol. 3 (Springer, New York 2004)

    Book  MATH  Google Scholar 

  138. K. Falconer: Fractal Geometry: Mathematical Foundations and Applications (Wiley, Hoboken 2003)

    Book  MATH  Google Scholar 

  139. T. Sauer, J.A. Yorke, M. Casdagli: Embedology, J. Stat. Phys. 65(3), 579–616 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  140. A. Wolf, J.B. Swift, H.L. Swinney, J.A. Vastano: Determining Lyapunov exponents from a time series, Phys. D 16(3), 285–317 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  141. L.D. Iasemidis, J.C. Sackellares, H.P. Zaveri, W.J. Williams: Phase space topography and the Lyapunov exponent of electrocorticograms in partial seizures, Brain Topogr. 2(3), 187–201 (1990)

    Article  Google Scholar 

  142. S. Micheloyannis, N. Flitzanis, E. Papanikolaou, M. Bourkas, D. Terzakis, S. Arvanitis, C.J. Stam: Usefulness of non-linear EEG analysis, Acta Neurol. Scand. 97(1), 13–19 (2009)

    Article  Google Scholar 

  143. W.J. Freeman: A field-theoretic approach to understanding scale-free neocortical dynamics, Biol. Cybern. 92(6), 350–359 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  144. W.J. Freeman, H. Erwin: Freeman k-set, Scholarpedia 3(2), 3238 (2008)

    Article  Google Scholar 

  145. R. Kozma, W. Freeman: Basic principles of the KIV model and its application to the navigation problem, Integr. Neurosci. 2(1), 125–145 (2003)

    Article  Google Scholar 

  146. R. Kozma, W.J. Freeman: The KIV model of intentional dynamics and decision making, Neural Netw. 22(3), 277–285 (2009)

    Article  MathSciNet  Google Scholar 

  147. H.-J. Chang, W.J. Freeman, B.C. Burke: Biologically modeled noise stabilizing neurodynamics for pattern recognition, Int. J. Bifurc. Chaos 8(2), 321–345 (1998)

    Article  MATH  Google Scholar 

  148. R. Kozma, J.W. Freeman: Chaotic resonance - methods and applications for robust classification of noisy and variable patterns, Int. J. Bifurc. Chaos 11(6), 1607–1629 (2001)

    Article  Google Scholar 

  149. R.J. McEliece, E.C. Posner, E. Rodemich, S. Venkatesh: The capacity of the Hopfield associative memory, IEEE Trans. Inf. Theory 33(4), 461–482 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  150. J. Ma: The asymptotic memory capacity of the generalized Hopfield network, Neural Netw. 12(9), 1207–1212 (1999)

    Article  Google Scholar 

  151. V. Gripon, C. Berrou: Sparse neural networks with large learning diversity, IEEE Trans. Neural Netw. 22(7), 1087–1096 (2011)

    Article  Google Scholar 

  152. I. Beliaev, R. Kozma: Studies on the memory capacity and robustness of chaotic dynamic neural networks, Int. Jt. Conf. Neural Netw., IEEE (2006) pp. 3991–3998

    Google Scholar 

  153. D.A. Leopold, N.K. Logothetis: Multistable phenomena: Changing views in perception, Trends Cogn. Sci. 3(7), 254–264 (1999)

    Article  Google Scholar 

  154. E.D. Lumer, K.J. Friston, G. Rees: Neural correlates of perceptual rivalry in the human brain, Science 280(5371), 1930–1934 (1998)

    Article  Google Scholar 

  155. G. Werner: Metastability, criticality and phase transitions in brain and its models, Biosystems 90(2), 496–508 (2007)

    Article  Google Scholar 

  156. W.J. Freeman: Understanding perception through neural codes, IEEE Trans. Biomed. Eng. 58(7), 1884–1890 (2011)

    Article  Google Scholar 

  157. R. Kozma, J.J. Davis, W.J. Freeman: Synchronized minima in ECoG power at frequencies between beta-gamma oscillations disclose cortical singularities in cognition, J. Neurosci. Neuroeng. 1(1), 13–23 (2012)

    Article  Google Scholar 

  158. R. Kozma: Neuropercolation, Scholarpedia 2(8), 1360 (2007)

    Article  MathSciNet  Google Scholar 

  159. E. Bullmore, O. Sporns: Complex brain networks: Graph theoretical analysis of structural and functional systems, Nat. Rev. Neurosci. 10(3), 186–198 (2009)

    Article  Google Scholar 

  160. R. Kozma, M. Puljic, P. Balister, B. Bollobas, W. Freeman: Neuropercolation: A random cellular automata approach to spatio-temporal neurodynamics, Lect. Notes Comput. Sci. 3305, 435–443 (2004)

    Article  MATH  Google Scholar 

  161. P. Balister, B. Bollobás, R. Kozma: Large deviations for mean field models of probabilistic cellular automata, Random Struct. Algorithm. 29(3), 399–415 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  162. R. Kozma, M. Puljic, L. Perlovsky: Modeling goal-oriented decision making through cognitive phase transitions, New Math. Nat. Comput. 5(1), 143–157 (2009)

    Article  MATH  Google Scholar 

  163. M. Puljic, R. Kozma: Broad-band oscillations by probabilistic cellular automata, J. Cell. Autom. 5(6), 491–507 (2010)

    MathSciNet  MATH  Google Scholar 

  164. S. Jo, T. Chang, I. Ebong, B. Bhadviya, P. Mazumder, W. Lu: Nanoscale memristor device as synapse in neuromorphic systems, Nano Lett. 10, 1297–1301 (2010)

    Article  Google Scholar 

  165. L. Smith: Handbook of Nature-Inspired and Innovative Computing: Integrating Classical Models with Emerging Technologies (Springer, New York 2006) pp. 433–475

    Book  Google Scholar 

  166. G. Indiveri, E. Chicca, R. Douglas: A VLSI array of low-power spiking neurons and bistable synapses with spike-timing dependent plasticity, IEEE Trans. Neural Netw. 17, 211–221 (2006)

    Article  Google Scholar 

  167. Editors of Scientific American: The Scientific American Book of the Brain (Scientifc American, New York 1999)

    Google Scholar 

  168. L. Chua, L. Yang: Cellular neural networks, Theory. IEEE Trans. Circuits Syst. CAS-35, 1257–1272 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  169. C. Zheng, H. Zhang, Z. Wang: Improved robust stability criteria for delayed cellular neural networks via the LMI approach, IEEE Trans. Circuits Syst. II – Expr. Briefs 57, 41–45 (2010)

    Article  Google Scholar 

  170. L. Chua: Memristor - The missing circuit element, IEEE Trans. Circuits Theory CT-18, 507–519 (1971)

    Article  Google Scholar 

  171. D. Strukov, G. Snider, D. Stewart, R. Williams: The missing memristor found, Nature 453, 80–83 (2008)

    Article  Google Scholar 

  172. Q. Xia, W. Robinett, M. Cumbie, N. Banerjee, T. Cardinali, J. Yang, W. Wu, X. Li, W. Tong, D. Strukov, G. Snider, G. Medeiros-Ribeiro, R. Williams: Memristor - CMOS hybrid integrated circuits for reconfigurable logic, Nano Lett. 9, 3640–3645 (2009)

    Article  Google Scholar 

  173. X. Wang, Y. Chen, H. Xi, H. Li, D. Dimitrov: Spintronic memristor through spin-torque-induced magnetization motion, IEEE Electron Device Lett. 30, 294–297 (2009)

    Article  Google Scholar 

  174. Y. Joglekar, S. Wolf: The elusive memristor: Properties of basic electrical circuits, Eur. J. Phys. 30, 661–675 (2009)

    Article  MATH  Google Scholar 

  175. M. Pickett, D. Strukov, J. Borghetti, J. Yang, G. Snider, D. Stewart, R. Williams: Switching dynamics in titanium dioxide memristive devices, J. Appl. Phys. 106(6), 074508 (2009)

    Article  Google Scholar 

  176. S. Adhikari, C. Yang, H. Kim, L. Chua: Memristor bridge synapse-based neural network and its learning, IEEE Trans. Neural Netw. Learn. Syst. 23(9), 1426–1435 (2012)

    Article  Google Scholar 

  177. B. Linares-Barranco, T. Serrano-Gotarredona: Exploiting memristive in adaptive spiking neuromorphic nanotechnology systems, 9th IEEE Conf. Nanotechnol., Genoa (2009) pp. 601–604

    Google Scholar 

  178. M. Holler, S. Tam, H. Castro, R. Benson: An electrically trainable artificial neural network (ETANN) with 10240 Floating gate synapsess, Int. J. Conf. Neural Netw., Washington (1989) pp. 191–196

    Chapter  Google Scholar 

  179. H. Withagen: Implementing backpropagation with analog hardware, Proc. IEEE ICNN-94, Orlando (1994) pp. 2015–2017

    Google Scholar 

  180. S. Lindsey, T. Lindblad: Survey of neural network hardware invited paper, Proc. SPIE Appl. Sci. Artif. Neural Netw. Conf., Orlando (1995) pp. 1194–1205

    Chapter  Google Scholar 

  181. H. Kim, M. Pd Sah, C. Yang, T. Roska, L. Chua: Neural synaptic weighting with a pulse-based memristor circuit, IEEE Trans. Circuits Syst. I 59(1), 148–158 (2012)

    Article  MathSciNet  Google Scholar 

  182. H. Kim, M. Pd Sah, C. Yang, T. Roska, L. Chua: Memristor bridge synapses, Proc. IEEE 100(6), 2061–2070 (2012)

    Article  Google Scholar 

  183. E. Lehtonen, M. Laiho: CNN using memristors for neighborhood connections, 12th Int. Workshop Cell. Nanoscale Netw. Appl. (CNNA), Berkeley (2010)

    Google Scholar 

  184. F. Merrikh-Bayat, F. Merrikh-Bayat, S. Shouraki: The neuro-fuzzy computing system with the capacity of implementation on a memristor crossbar and optimization-free hardware training, IEEE Trans. Fuzzy Syst. 22(5), 1272–1287 (2014)

    Article  Google Scholar 

  185. G. Snider: Spike-timing-dependent learning in memristive nanodevices, IEEE Int. Symp. Nanoscale Archit., Anaheim (2008) pp. 85–92

    Google Scholar 

  186. I. Ebong, D. Deshpande, Y. Yilmaz, P. Mazumder: Multi-purpose neuro-architecture with memristors, 11th IEEE Conf. Nanotechnol., Portland, Oregon (2011) pp. 1194–1205

    Google Scholar 

  187. H. Manem, J. Rajendran, G. Rose: Stochastic gradient descent inspired training technique for a CMOS/Nano memristive trainable threshold gate way, IEEE Trans. Circuits Syst. I 59(5), 1051–1060 (2012)

    Article  MathSciNet  Google Scholar 

  188. G. Howard, E. Gale, L. Bull, B. Costello, A. Adamatzky: Evolution of plastic learning in spiking networks via memristive connections, IEEE Trans. Evol. Comput. 16(5), 711–729 (2012)

    Article  Google Scholar 

  189. S. Wen, Z. Zeng, T. Huang: Exponential stability analysis of memristor-based recurrent neural networks with time-varying delays, Neurocomputing 97(15), 233–240 (2012)

    Article  Google Scholar 

  190. A. Wu, Z. Zeng: Dynamics behaviors of memristor-based recurrent neural networks with time-varying delays, Neural Netw. 36, 1–10 (2012)

    Article  MATH  Google Scholar 

  191. A. Cichocki, R. Unbehauen: Neural Networks for Optimization and Signal Processing (Wiley, New York 1993)

    MATH  Google Scholar 

  192. J. Wang: Recurrent neural networks for optimization. In: Fuzzy Logic and Neural Network Handbook, ed. by C.H. Chen (McGraw-Hill, New York 1996), pp. 4.1–4.35

    Google Scholar 

  193. Y. Xia, J. Wang: Recurrent neural networks for optimization: The state of the art. In: Recurrent Neural Networks: Design and Applications, ed. by L.R. Medsker, L.C. Jain (CRC, Boca Raton 1999), 13–45

    Google Scholar 

  194. Q. Liu, J. Wang: Recurrent neural networks with discontinuous activation functions for convex optimization. In: Integration of Swarm Intelligence and Artifical Neutral Network, ed. by S. Dehuri, S. Ghosh, S.B. Cho (World Scientific, Singapore 2011), 95–119

    Google Scholar 

  195. I.B. Pyne: Linear programming on an electronic analogue computer, Trans. Am. Inst. Elect. Eng. 75(2), 139–143 (1956)

    Google Scholar 

  196. L.O. Chua, G. Lin: Nonlinear programming without computation, IEEE Trans. Circuits Syst. 31(2), 182–189 (1984)

    Article  MathSciNet  Google Scholar 

  197. G. Wilson: Quadratic programming analogs, IEEE Trans. Circuits Syst. 33(9), 907–911 (1986)

    Article  MATH  Google Scholar 

  198. J.J. Hopfield, D.W. Tank: Neural computation of decisions in optimization problems, Biol. Cybern. 52(3), 141–152 (1985)

    MathSciNet  MATH  Google Scholar 

  199. J.J. Hopfield, D.W. Tank: Computing with neural circuits -- a model, Science 233(4764), 625–633 (1986)

    Article  Google Scholar 

  200. D.W. Tank, J.J. Hopfield: Simple neural optimization networks: an A/D converter, signal decision circuit, and a linear programming circuit, IEEE Trans. Circuits Syst. 33(5), 533–541 (1986)

    Article  Google Scholar 

  201. M.P. Kennedy, L.O. Chua: Neural networks for nonlinear programming, IEEE Trans. Circuits Syst. 35(5), 554–562 (1988)

    Article  MathSciNet  Google Scholar 

  202. A. Rodriguez-Vazquez, R. Dominguez-Castro, A. Rueda, J.L. Huertas, E. Sanchez-Sinencio: Nonlinear switch-capacitor neural networks for optimization problems, IEEE Trans. Circuits Syst. 37(3), 384–398 (1990)

    Article  MathSciNet  Google Scholar 

  203. S. Sudharsanan, M. Sundareshan: Exponential stability and a systematic synthesis of a neural network for quadratic minimization, Neural Netw. 4, 599–613 (1991)

    Article  Google Scholar 

  204. S. Zhang, A.G. Constantinides: Lagrange programming neural network, IEEE Trans. Circuits Syst. 39(7), 441–452 (1992)

    Article  MATH  Google Scholar 

  205. S. Zhang, X. Zhu, L. Zou: Second-order neural nets for constrained optimization, IEEE Trans. Neural Netw. 3(6), 1021–1024 (1992)

    Article  Google Scholar 

  206. A. Bouzerdoum, T.R. Pattison: Neural network for quadratic optimization with bound constraints, IEEE Trans. Neural Netw. 4(2), 293–304 (1993)

    Article  Google Scholar 

  207. M. Ohlsson, C. Peterson, B. Soderberg: Neural networks for optimization problems with inequality constraints: The knapsack problem, Neural Comput. 5, 331–339 (1993)

    Article  Google Scholar 

  208. J. Wang: Analysis and design of a recurrent neural network for linear programming, IEEE Trans. Circuits Syst. I 40(9), 613–618 (1993)

    Article  MATH  Google Scholar 

  209. W.E. Lillo, M.H. Loh, S. Hui, S.H. Zak: On solving constrained optimization problems with neural networks: A penalty method approach, IEEE Trans. Neural Netw. 4(6), 931–940 (1993)

    Article  Google Scholar 

  210. J. Wang: A deterministic annealing neural network for convex programming, Neural Netw. 7(4), 629–641 (1994)

    Article  MATH  Google Scholar 

  211. S.H. Zak, V. Upatising, S. Hui: Solving linear programming problems with neural networks: A comparative study, IEEE Trans. Neural Netw. 6, 94–104 (1995)

    Article  Google Scholar 

  212. Y. Xia, J. Wang: Neural network for solving linear programming problems with bounded variables, IEEE Trans. Neural Netw. 6(2), 515–519 (1995)

    Article  Google Scholar 

  213. M. Vidyasagar: Minimum-seeking properties of analog neural networks with multilinear objective functions, IEEE Trans. Autom. Control 40(8), 1359–1375 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  214. M. Forti, A. Tesi: New conditions for global stability of neural networks with application to linear and quadratic programming problems, IEEE Trans. Circuits Syst. I 42(7), 354–366 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  215. A. Cichocki, R. Unbehauen, K. Weinzierl, R. Holzel: A new neural network for solving linear programming problems, Eur. J. Oper. Res. 93, 244–256 (1996)

    Article  MATH  Google Scholar 

  216. Y. Xia: A new neural network for solving linear programming problems and its application, IEEE Trans. Neural Netw. 7(2), 525–529 (1996)

    Article  Google Scholar 

  217. X. Wu, Y. Xia, J. Li, W.K. Chen: A high-performance neural network for solving linear and quadratic programming problems, IEEE Trans. Neural Netw. 7(3), 1996 (1996)

    Google Scholar 

  218. Y. Xia: A new neural network for solving linear and quadratic programming problems, IEEE Trans. Neural Netw. 7(6), 1544–1547 (1996)

    Article  Google Scholar 

  219. Y. Xia: Neural network for solving extended linear programming problems, IEEE Trans. Neural Netw. 8(3), 803–806 (1997)

    Article  Google Scholar 

  220. M.J. Perez-Ilzarbe: Convergence analysis of a discrete-time recurrent neural network to perform quadratic real optimization with bound constraints, IEEE Trans. Neural Netw. 9(6), 1344–1351 (1998)

    Article  Google Scholar 

  221. M.C.M. Teixeira, S.H. Zak: Analog neural nonderivative optimizers, IEEE Trans. Neural Netw. 9(4), 629–638 (1998)

    Article  Google Scholar 

  222. Y. Xia, J. Wang: A general methodology for designing globally convergent optimization neural networks, IEEE Trans. Neural Netw. 9(6), 1331–1343 (1998)

    Article  Google Scholar 

  223. E. Chong, S. Hui, H. Zak: An analysis of a class of neural networks for solving linear programming problems, IEEE Trans. Autom. Control 44(11), 1995–2006 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  224. Y. Xia, J. Wang: Global exponential stability of recurrent neural networks for solving optimization and related problems, IEEE Trans. Neural Netw. 11(4), 1017–1022 (2000)

    Article  Google Scholar 

  225. X. Liang, J. Wang: A recurrent neural network for nonlinear optimization with a continuously differentiable objective function and bound constraints, IEEE Trans. Neural Netw. 11(6), 1251–1262 (2000)

    Article  Google Scholar 

  226. Y. Leung, K. Chen, Y. Jiao, X. Gao, K. Leung: A new gradient-based neural network for solving linear and quadratic programming problems, IEEE Trans. Neural Netw. 12(5), 1074–1083 (2001)

    Article  Google Scholar 

  227. X. Liang: A recurrent neural network for nonlinear continuously differentiable optimization over a compact convex subset, IEEE Trans. Neural Netw. 12(6), 1487–1490 (2001)

    Article  Google Scholar 

  228. Y. Xia, H. Leung, J. Wang: A projection neural network and its application to constrained optimization problems, IEEE Trans. Circuits Syst. I 49(4), 447–458 (2002)

    Article  MathSciNet  Google Scholar 

  229. R. Fantacci, M. Forti, M. Marini, D. Tarchi, G. Vannuccini: A neural network for constrained optimization with application to CDMA communication systems, IEEE Trans. Circuits Syst. II 50(8), 484–487 (2003)

    Article  Google Scholar 

  230. Y. Leung, K. Chen, X. Gao: A high-performance feedback neural network for solving convex nonlinear programming problems, IEEE Trans. Neural Netw. 14(6), 1469–1477 (2003)

    Article  Google Scholar 

  231. Y. Xia, J. Wang: A general projection neural network for solving optimization and related problems, IEEE Trans. Neural Netw. 15, 318–328 (2004)

    Article  MathSciNet  Google Scholar 

  232. X. Gao: A novel neural network for nonlinear convex programming, IEEE Trans. Neural Netw. 15(3), 613–621 (2004)

    Article  Google Scholar 

  233. X. Gao, L. Liao, W. Xue: A neural network for a class of convex quadratic minimax problems with constraints, IEEE Trans. Neural Netw. 15(3), 622–628 (2004)

    Article  Google Scholar 

  234. Y. Xia, J. Wang: A recurrent neural network for nonlinear convex optimization subject to nonlinear inequality constraints, IEEE Trans. Circuits Syst. I 51(7), 1385–1394 (2004)

    Article  MathSciNet  Google Scholar 

  235. M. Forti, P. Nistri, M. Quincampoix: Generalized neural network for nonsmooth nonlinear programming problems, IEEE Trans. Circuits Syst. I 51(9), 1741–1754 (2004)

    Article  MathSciNet  Google Scholar 

  236. Y. Xia, G. Feng, J. Wang: A recurrent neural network with exponential convergence for solving convex quadratic program and linear piecewise equations, Neural Netw. 17(7), 1003–1015 (2004)

    Article  MATH  Google Scholar 

  237. Y. Xia, J. Wang: Recurrent neural networks for solving nonlinear convex programs with linear constraints, IEEE Trans. Neural Netw. 16(2), 379–386 (2005)

    Article  Google Scholar 

  238. Q. Liu, J. Cao, Y. Xia: A delayed neural network for solving linear projection equations and its applications, IEEE Trans. Neural Netw. 16(4), 834–843 (2005)

    Article  Google Scholar 

  239. X. Hu, J. Wang: Solving pseudomonotone variational inequalities and pseudoconvex optimization problems using the projection neural network, IEEE Trans. Neural Netw. 17(6), 1487–1499 (2006)

    Article  Google Scholar 

  240. S. Liu, J. Wang: A simplified dual neural network for quadratic programming with its KWTA application, IEEE Trans. Neural Netw. 17(6), 1500–1510 (2006)

    Article  Google Scholar 

  241. Y. Yang, J. Cao: Solving quadratic programming problems by delayed projection neural network, IEEE Trans. Neural Netw. 17(6), 1630–1634 (2006)

    Article  Google Scholar 

  242. X. Hu, J. Wang: Design of general projection neural network for solving monotone linear variational inequalities and linear and quadratic optimization problems, IEEE Trans. Syst. Man Cybern. B 37(5), 1414–1421 (2007)

    Article  Google Scholar 

  243. Q. Liu, J. Wang: A one-layer recurrent neural network with a discontinuous hard-limiting activation function for quadratic programming, IEEE Trans. Neural Netw. 19(4), 558–570 (2008)

    Article  Google Scholar 

  244. Q. Liu, J. Wang: A one-layer recurrent neural network with a discontinuous activation function for linear programming, Neural Comput. 20(5), 1366–1383 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  245. Y. Xia, G. Feng, J. Wang: A novel neural network for solving nonlinear optimization problems with inequality constraints, IEEE Trans. Neural Netw. 19(8), 1340–1353 (2008)

    Article  Google Scholar 

  246. M.P. Barbarosou, N.G. Maratos: A nonfeasible gradient projection recurrent neural network for equality-constrained optimization problems, IEEE Trans. Neural Netw. 19(10), 1665–1677 (2008)

    Article  Google Scholar 

  247. X. Hu, J. Wang: An improved dual neural network for solving a class of quadratic programming problems and its k-winners-take-all application, IEEE Trans. Neural Netw. 19(12), 2022–2031 (2008)

    Article  Google Scholar 

  248. X. Xue, W. Bian: Subgradient-based neural networks for nonsmooth convex optimization problems, IEEE Trans. Circuits Syst. I 55(8), 2378–2391 (2008)

    Article  MathSciNet  Google Scholar 

  249. W. Bian, X. Xue: Subgradient-based neural networks for nonsmooth nonconvex optimization problems, IEEE Trans. Neural Netw. 20(6), 1024–1038 (2009)

    Article  Google Scholar 

  250. X. Hu, C. Sun, B. Zhang: Design of recurrent neural networks for solving constrained least absolute deviation problems, IEEE Trans. Neural Netw. 21(7), 1073–1086 (2010)

    Article  Google Scholar 

  251. Q. Liu, J. Wang: Finite-time convergent recurrent neural network with a hard-limiting activation function for constrained optimization with piecewise-linear objective functions, IEEE Trans. Neural Netw. 22(4), 601–613 (2011)

    Article  Google Scholar 

  252. Q. Liu, J. Wang: A one-layer recurrent neural network for constrained nonsmooth optimization, IEEE Trans. Syst. Man Cybern. 40(5), 1323–1333 (2011)

    Google Scholar 

  253. L. Cheng, Z. Hou, Y. Lin, M. Tan, W.C. Zhang, F. Wu: Recurrent neural network for nonsmooth convex optimization problems with applications to the identification of genetic regulatory networks, IEEE Trans. Neural Netw. 22(5), 714–726 (2011)

    Article  Google Scholar 

  254. Z. Guo, Q. Liu, J. Wang: A one-layer recurrent neural network for pseudoconvex optimization subject to linear equality constraints, IEEE Trans. Neural Netw. 22(12), 1892–1900 (2011)

    Article  MathSciNet  Google Scholar 

  255. Q. Liu, Z. Guo, J. Wang: A one-layer recurrent neural network for constrained pseudoconvex optimization and its application for dynamic portfolio optimization, Neural Netw. 26(1), 99–109 (2012)

    Article  MATH  Google Scholar 

  256. W. Bian, X. Chen: Smoothing neural network for constrained non-Lipschitz optimization with applications, IEEE Trans. Neural Netw. Learn. Syst. 23(3), 399–411 (2012)

    Article  Google Scholar 

  257. Y. Xia: An extended projection neural network for constrained optimization, Neural Comput. 16(4), 863–883 (2004)

    Article  MATH  Google Scholar 

  258. J. Wang, Y. Xia: Analysis and design of primal-dual assignment networks, IEEE Trans. Neural Netw. 9(1), 183–194 (1998)

    Article  MathSciNet  Google Scholar 

  259. Y. Xia, G. Feng, J. Wang: A primal-dual neural network for online resolving constrained kinematic redundancy in robot motion control, IEEE Trans. Syst. Man Cybern. B 35(1), 54–64 (2005)

    Article  Google Scholar 

  260. Y. Xia, J. Wang: A dual neural network for kinematic control of redundant robot manipulators, IEEE Trans. Syst. Man Cybern. B 31(1), 147–154 (2001)

    Article  Google Scholar 

  261. Y. Zhang, J. Wang: A dual neural network for constrained joint torque optimization of kinematically redundant manipulators, IEEE Trans. Syst. Man Cybern. B 32(5), 654–662 (2002)

    Article  Google Scholar 

  262. Y. Zhang, J. Wang, Y. Xu: A dual neural network for bi-criteria kinematic control redundant manipulators, IEEE Trans. Robot. Autom. 18(6), 923–931 (2002)

    Article  Google Scholar 

  263. Y. Zhang, J. Wang, Y. Xia: A dual neural network for redundancy resolution of kinematically redundant manipulators subject to joint limits and joint velocity limits, IEEE Trans. Neural Netw. 14(3), 658–667 (2003)

    Article  Google Scholar 

  264. A. Cichocki, R. Unbehauen: Neural networks for solving systems of linear equations and related problems, IEEE Trans. Circuits Syst. I 39(2), 124–138 (1992)

    Article  MATH  Google Scholar 

  265. A. Cichocki, R. Unbehauen: Neural networks for solving systems of linear equations – part II: Minimax and least absolute value problems, IEEE Trans. Circuits Syst. II 39(9), 619–633 (1992)

    Article  MATH  Google Scholar 

  266. J. Wang: Recurrent neural networks for computing pseudoinverse of rank-deficient matrices, SIAM J. Sci. Comput. 18(5), 1479–1493 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  267. G.G. Lendaris, K. Mathia, R. Saeks: Linear Hopfield networks and constrained optimization, IEEE Trans. Syst. Man Cybern. B 29(1), 114–118 (1999)

    Article  Google Scholar 

  268. Y. Xia, J. Wang, D.L. Hung: Recurrent neural networks for solving linear inequalities and equations, IEEE Trans. Circuits Syst. I 46(4), 452–462 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  269. J. Wang: A recurrent neural network for solving the shortest path problem, IEEE Trans. Circuits Syst. I 43(6), 482–486 (1996)

    Article  Google Scholar 

  270. J. Wang: Primal and dual neural networks for shortest-path routing, IEEE Trans. Syst. Man Cybern. A 28(6), 864–869 (1998)

    Article  Google Scholar 

  271. Y. Xia, J. Wang: A discrete-time recurrent neural network for shortest-path routing, IEEE Trans. Autom. Control 45(11), 2129–2134 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  272. D. Anguita, A. Boni: Improved neural network for SVM learning, IEEE Trans. Neural Netw. 13(5), 1243–1244 (2002)

    Article  Google Scholar 

  273. Y. Xia, J. Wang: A one-layer recurrent neural network for support vector machine learning, IEEE Trans. Syst. Man Cybern. B 34(2), 1261–1269 (2004)

    Article  Google Scholar 

  274. L.V. Ferreira, E. Kaszkurewicz, A. Bhaya: Support vector classifiers via gradient systems with discontinuous right-hand sides, Neural Netw. 19(10), 1612–1623 (2006)

    Article  MATH  Google Scholar 

  275. J. Wang: Analysis and design of an analog sorting network, IEEE Trans. Neural Netw. 6, 962–971 (1995)

    Article  Google Scholar 

  276. B. Apolloni, I. Zoppis: Subsymbolically managing pieces of symbolical functions for sorting, IEEE Trans. Neural Netw. 10(5), 1099–1122 (1999)

    Article  Google Scholar 

  277. J. Wang: Analysis and design of k-winners-take-all model with a single state variable and Heaviside step activation function, IEEE Trans. Neural Netw. 21(9), 1496–1506 (2010)

    Article  Google Scholar 

  278. Q. Liu, J. Wang: Two k-winners-take-all networks with discontinuous activation functions, Neural Netw. 21, 406–413 (2008)

    Article  MATH  Google Scholar 

  279. Y. Xia, M. S. Kamel: Cooperative learning algorithms for data fusion using novel L1 estimation, IEEE Trans. Signal Process. 56(3), 1083–-1095 (2008)

    Google Scholar 

  280. B. Baykal, A.G. Constantinides: A neural approach to the underdetermined-order recursive least-squares adaptive filtering, Neural Netw. 10(8), 1523–1531 (1997)

    Article  MATH  Google Scholar 

  281. Y. Sun: Hopfield neural network based algorithms for image restoration and reconstruction – Part I: Algorithms and simulations, IEEE Trans. Signal Process. 49(7), 2105–2118 (2000)

    Article  MATH  Google Scholar 

  282. X.Z. Wang, J.Y. Cheung, Y.S. Xia, J.D.Z. Chen: Minimum fuel neural networks and their applications to overcomplete signal representations, IEEE Trans. Circuits Syst. I 47(8), 1146–1159 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  283. X.Z. Wang, J.Y. Cheung, Y.S. Xia, J.D.Z. Chen: Neural implementation of unconstrained minimum L1-norm optimization—least absolute deviation model and its application to time delay estimation, IEEE Trans. Circuits Syst. II 47(11), 1214–1226 (2000)

    Article  MathSciNet  Google Scholar 

  284. P.-R. Chang, W.-H. Yang, K.-K. Chan: A neural network approach to MVDR beamforming problem, IEEE Trans. Antennas Propag. 40(3), 313–322 (1992)

    Article  Google Scholar 

  285. Y. Xia, G.G. Feng: A neural network for robust LCMP beamforming, Signal Process. 86(3), 2901–2912 (2006)

    Article  MATH  Google Scholar 

  286. J. Wang, G. Wu: A multilayer recurrent neural network for on-line synthesis of minimum-norm linear feedback control systems via pole assignment, Automatica 32(3), 435–442 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  287. Y. Zhang, J. Wang: Global exponential stability of recurrent neural networks for synthesizing linear feedback control systems via pole assignment, IEEE Trans. Neural Netw. 13(3), 633–644 (2002)

    Article  Google Scholar 

  288. Y. Zhang, J. Wang: Recurrent neural networks for nonlinear output regulation, Automatica 37(8), 1161–1173 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  289. S. Hu, J. Wang: Multilayer recurrent neural networks for online robust pole assignment, IEEE Trans. Circuits Syst. I 50(11), 1488–1494 (2003)

    Article  MathSciNet  Google Scholar 

  290. Y. Pan, J. Wang: Model predictive control of unknown nonlinear dynamical systems based on recurrent neural networks, IEEE Trans. Ind. Electron. 59(8), 3089–3101 (2012)

    Article  Google Scholar 

  291. Z. Yan, J. Wang: Model predictive control of nonlinear systems with unmodeled dynamics based on feedforward and recurrent neural networks, IEEE Trans. Ind. Inf. 8(4), 746–756 (2012)

    Article  Google Scholar 

  292. Z. Yan, J. Wang: Model predictive control of tracking of underactuated vessels based on recurrent neural networks, IEEE J. Ocean. Eng. 37(4), 717–726 (2012)

    Article  Google Scholar 

  293. J. Wang, Q. Hu, D. Jiang: A Lagrangian network for kinematic control of redundant robot manipulators, IEEE Trans. Neural Netw. 10(5), 1123–1132 (1999)

    Article  Google Scholar 

  294. H. Ding, S.K. Tso: A fully neural-network-based planning scheme for torque minimization of redundant manipulators, IEEE Trans. Ind. Electron. 46(1), 199–206 (1999)

    Article  Google Scholar 

  295. H. Ding, J. Wang: Recurrent neural networks for minimum infinity-norm kinematic control of redundant manipulators, IEEE Trans. Syst. Man Cybern. A 29(3), 269–276 (1999)

    Article  Google Scholar 

  296. W.S. Tang, J. Wang: Two recurrent neural networks for local joint torque optimization of kinematically redundant manipulators, IEEE Trans. Syst. Man Cybern. B 30(1), 120–128 (2000)

    Article  Google Scholar 

  297. W.S. Tang, J. Wang: A recurrent neural network for minimum infinity-norm kinematic control of redundant manipulators with an improved problem formulation and reduced architectural complexity, IEEE Trans. Syst. Man Cybern. B 31(1), 98–105 (2001)

    Article  Google Scholar 

  298. Y. Zhang, J. Wang: Obstacle avoidance for kinematically redundant manipulators using a dual neural network, IEEE Trans. Syst. Man Cybern. B 4(1), 752–759 (2004)

    Article  Google Scholar 

  299. Y. Xia, J. Wang, L.-M. Fok: Grasping force optimization of multi-fingered robotic hands using a recurrent neural network, IEEE Trans. Robot. Autom. 20(3), 549–554 (2004)

    Article  Google Scholar 

  300. Q. Liu, C. Dang, T. Huang: A one-layer recurrent neural network for real-time portfolio optimization with probability criterion, IEEE Trans. Cybern. 43(1), 14–23 (2013)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dep. Mechanical & Automation Engineering, The Chinese University of Hong Kong, Shatin, New Territories, Hongkong, Hong Kong

    Jun Wang

  2. Dep. Mathematical Sciences, University of Memphis, TN 38152, Memphis, USA

    Robert Kozma

  3. Dep. Control Science and Engineering, Huazhong University of Science and Technology, No. 1037, Luoyu Road, 430074, Wuhan, China

    Zhigang Zeng

Authors
  1. Robert Kozma
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jun Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zhigang Zeng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Robert Kozma .

Editor information

Editors and Affiliations

  1. Systems Research Inst., Polish Academy of Sciences, ul. Newelska 6, 01-447, Warsaw, Poland

    Janusz Kacprzyk

  2. Dep. Electrical and Computer Engineering, University of Alberta, 116 Street 9107, T6J 2V4, Edmonton, Alberta, Canada

    Witold Pedrycz

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Kozma, R., Wang, J., Zeng, Z. (2015). Neurodynamics. In: Kacprzyk, J., Pedrycz, W. (eds) Springer Handbook of Computational Intelligence. Springer Handbooks. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-43505-2_33

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-43505-2_33

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-43504-5

  • Online ISBN: 978-3-662-43505-2

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation